/*
 * Copyright (c) 1996, 2013, Oracle and/or its affiliates. All rights reserved.
 * ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
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 *
 *
 */

/*
 * Portions Copyright (c) 1995  Colin Plumb.  All rights reserved.
 */

package java.math;

import java.io.IOException;
import java.io.ObjectInputStream;
import java.io.ObjectOutputStream;
import java.io.ObjectStreamField;
import java.util.Arrays;
import java.util.Random;
import java.util.concurrent.ThreadLocalRandom;
import sun.misc.DoubleConsts;
import sun.misc.FloatConsts;

/**
 * Immutable arbitrary-precision integers.  All operations behave as if
 * BigIntegers were represented in two's-complement notation (like Java's
 * primitive integer types).  BigInteger provides analogues to all of Java's
 * primitive integer operators, and all relevant methods from java.lang.Math.
 * Additionally, BigInteger provides operations for modular arithmetic, GCD
 * calculation, primality testing, prime generation, bit manipulation,
 * and a few other miscellaneous operations.
 *
 * <p>Semantics of arithmetic operations exactly mimic those of Java's integer
 * arithmetic operators, as defined in <i>The Java Language Specification</i>.
 * For example, division by zero throws an {@code ArithmeticException}, and
 * division of a negative by a positive yields a negative (or zero) remainder.
 * All of the details in the Spec concerning overflow are ignored, as
 * BigIntegers are made as large as necessary to accommodate the results of an
 * operation.
 *
 * <p>Semantics of shift operations extend those of Java's shift operators
 * to allow for negative shift distances.  A right-shift with a negative
 * shift distance results in a left shift, and vice-versa.  The unsigned
 * right shift operator ({@code >>>}) is omitted, as this operation makes
 * little sense in combination with the "infinite word size" abstraction
 * provided by this class.
 *
 * <p>Semantics of bitwise logical operations exactly mimic those of Java's
 * bitwise integer operators.  The binary operators ({@code and},
 * {@code or}, {@code xor}) implicitly perform sign extension on the shorter
 * of the two operands prior to performing the operation.
 *
 * <p>Comparison operations perform signed integer comparisons, analogous to
 * those performed by Java's relational and equality operators.
 *
 * <p>Modular arithmetic operations are provided to compute residues, perform
 * exponentiation, and compute multiplicative inverses.  These methods always
 * return a non-negative result, between {@code 0} and {@code (modulus - 1)},
 * inclusive.
 *
 * <p>Bit operations operate on a single bit of the two's-complement
 * representation of their operand.  If necessary, the operand is sign-
 * extended so that it contains the designated bit.  None of the single-bit
 * operations can produce a BigInteger with a different sign from the
 * BigInteger being operated on, as they affect only a single bit, and the
 * "infinite word size" abstraction provided by this class ensures that there
 * are infinitely many "virtual sign bits" preceding each BigInteger.
 *
 * <p>For the sake of brevity and clarity, pseudo-code is used throughout the
 * descriptions of BigInteger methods.  The pseudo-code expression
 * {@code (i + j)} is shorthand for "a BigInteger whose value is
 * that of the BigInteger {@code i} plus that of the BigInteger {@code j}."
 * The pseudo-code expression {@code (i == j)} is shorthand for
 * "{@code true} if and only if the BigInteger {@code i} represents the same
 * value as the BigInteger {@code j}."  Other pseudo-code expressions are
 * interpreted similarly.
 *
 * <p>All methods and constructors in this class throw
 * {@code NullPointerException} when passed
 * a null object reference for any input parameter.
 *
 * BigInteger must support values in the range
 * -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive)
 * and may support values outside of that range.
 *
 * The range of probable prime values is limited and may be less than
 * the full supported positive range of {@code BigInteger}.
 * The range must be at least 1 to 2<sup>500000000</sup>.
 *
 * @author Josh Bloch
 * @author Michael McCloskey
 * @author Alan Eliasen
 * @author Timothy Buktu
 * @implNote BigInteger constructors and operations throw {@code ArithmeticException} when the
 * result is out of the supported range of -2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive) to
 * +2<sup>{@code Integer.MAX_VALUE}</sup> (exclusive).
 * @see BigDecimal
 * @since JDK1.1
 */

public class BigInteger extends Number implements Comparable<BigInteger> {

  /**
   * The signum of this BigInteger: -1 for negative, 0 for zero, or
   * 1 for positive.  Note that the BigInteger zero <i>must</i> have
   * a signum of 0.  This is necessary to ensures that there is exactly one
   * representation for each BigInteger value.
   *
   * @serial
   */
  final int signum;

  /**
   * The magnitude of this BigInteger, in <i>big-endian</i> order: the
   * zeroth element of this array is the most-significant int of the
   * magnitude.  The magnitude must be "minimal" in that the most-significant
   * int ({@code mag[0]}) must be non-zero.  This is necessary to
   * ensure that there is exactly one representation for each BigInteger
   * value.  Note that this implies that the BigInteger zero has a
   * zero-length mag array.
   */
  final int[] mag;

  // These "redundant fields" are initialized with recognizable nonsense
  // values, and cached the first time they are needed (or never, if they
  // aren't needed).

  /**
   * One plus the bitCount of this BigInteger. Zeros means unitialized.
   *
   * @serial
   * @see #bitCount
   * @deprecated Deprecated since logical value is offset from stored value and correction factor is
   * applied in accessor method.
   */
  @Deprecated
  private int bitCount;

  /**
   * One plus the bitLength of this BigInteger. Zeros means unitialized.
   * (either value is acceptable).
   *
   * @serial
   * @see #bitLength()
   * @deprecated Deprecated since logical value is offset from stored value and correction factor is
   * applied in accessor method.
   */
  @Deprecated
  private int bitLength;

  /**
   * Two plus the lowest set bit of this BigInteger, as returned by
   * getLowestSetBit().
   *
   * @serial
   * @see #getLowestSetBit
   * @deprecated Deprecated since logical value is offset from stored value and correction factor is
   * applied in accessor method.
   */
  @Deprecated
  private int lowestSetBit;

  /**
   * Two plus the index of the lowest-order int in the magnitude of this
   * BigInteger that contains a nonzero int, or -2 (either value is acceptable).
   * The least significant int has int-number 0, the next int in order of
   * increasing significance has int-number 1, and so forth.
   *
   * @deprecated Deprecated since logical value is offset from stored value and correction factor is
   * applied in accessor method.
   */
  @Deprecated
  private int firstNonzeroIntNum;

  /**
   * This mask is used to obtain the value of an int as if it were unsigned.
   */
  final static long LONG_MASK = 0xffffffffL;

  /**
   * This constant limits {@code mag.length} of BigIntegers to the supported
   * range.
   */
  private static final int MAX_MAG_LENGTH = Integer.MAX_VALUE / Integer.SIZE + 1; // (1 << 26)

  /**
   * Bit lengths larger than this constant can cause overflow in searchLen
   * calculation and in BitSieve.singleSearch method.
   */
  private static final int PRIME_SEARCH_BIT_LENGTH_LIMIT = 500000000;

  /**
   * The threshold value for using Karatsuba multiplication.  If the number
   * of ints in both mag arrays are greater than this number, then
   * Karatsuba multiplication will be used.   This value is found
   * experimentally to work well.
   */
  private static final int KARATSUBA_THRESHOLD = 80;

  /**
   * The threshold value for using 3-way Toom-Cook multiplication.
   * If the number of ints in each mag array is greater than the
   * Karatsuba threshold, and the number of ints in at least one of
   * the mag arrays is greater than this threshold, then Toom-Cook
   * multiplication will be used.
   */
  private static final int TOOM_COOK_THRESHOLD = 240;

  /**
   * The threshold value for using Karatsuba squaring.  If the number
   * of ints in the number are larger than this value,
   * Karatsuba squaring will be used.   This value is found
   * experimentally to work well.
   */
  private static final int KARATSUBA_SQUARE_THRESHOLD = 128;

  /**
   * The threshold value for using Toom-Cook squaring.  If the number
   * of ints in the number are larger than this value,
   * Toom-Cook squaring will be used.   This value is found
   * experimentally to work well.
   */
  private static final int TOOM_COOK_SQUARE_THRESHOLD = 216;

  /**
   * The threshold value for using Burnikel-Ziegler division.  If the number
   * of ints in the divisor are larger than this value, Burnikel-Ziegler
   * division may be used.  This value is found experimentally to work well.
   */
  static final int BURNIKEL_ZIEGLER_THRESHOLD = 80;

  /**
   * The offset value for using Burnikel-Ziegler division.  If the number
   * of ints in the divisor exceeds the Burnikel-Ziegler threshold, and the
   * number of ints in the dividend is greater than the number of ints in the
   * divisor plus this value, Burnikel-Ziegler division will be used.  This
   * value is found experimentally to work well.
   */
  static final int BURNIKEL_ZIEGLER_OFFSET = 40;

  /**
   * The threshold value for using Schoenhage recursive base conversion. If
   * the number of ints in the number are larger than this value,
   * the Schoenhage algorithm will be used.  In practice, it appears that the
   * Schoenhage routine is faster for any threshold down to 2, and is
   * relatively flat for thresholds between 2-25, so this choice may be
   * varied within this range for very small effect.
   */
  private static final int SCHOENHAGE_BASE_CONVERSION_THRESHOLD = 20;

  /**
   * The threshold value for using squaring code to perform multiplication
   * of a {@code BigInteger} instance by itself.  If the number of ints in
   * the number are larger than this value, {@code multiply(this)} will
   * return {@code square()}.
   */
  private static final int MULTIPLY_SQUARE_THRESHOLD = 20;

  // Constructors

  /**
   * Translates a byte array containing the two's-complement binary
   * representation of a BigInteger into a BigInteger.  The input array is
   * assumed to be in <i>big-endian</i> byte-order: the most significant
   * byte is in the zeroth element.
   *
   * @param val big-endian two's-complement binary representation of BigInteger.
   * @throws NumberFormatException {@code val} is zero bytes long.
   */
  public BigInteger(byte[] val) {
    if (val.length == 0) {
      throw new NumberFormatException("Zero length BigInteger");
    }

    if (val[0] < 0) {
      mag = makePositive(val);
      signum = -1;
    } else {
      mag = stripLeadingZeroBytes(val);
      signum = (mag.length == 0 ? 0 : 1);
    }
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  /**
   * This private constructor translates an int array containing the
   * two's-complement binary representation of a BigInteger into a
   * BigInteger. The input array is assumed to be in <i>big-endian</i>
   * int-order: the most significant int is in the zeroth element.
   */
  private BigInteger(int[] val) {
    if (val.length == 0) {
      throw new NumberFormatException("Zero length BigInteger");
    }

    if (val[0] < 0) {
      mag = makePositive(val);
      signum = -1;
    } else {
      mag = trustedStripLeadingZeroInts(val);
      signum = (mag.length == 0 ? 0 : 1);
    }
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  /**
   * Translates the sign-magnitude representation of a BigInteger into a
   * BigInteger.  The sign is represented as an integer signum value: -1 for
   * negative, 0 for zero, or 1 for positive.  The magnitude is a byte array
   * in <i>big-endian</i> byte-order: the most significant byte is in the
   * zeroth element.  A zero-length magnitude array is permissible, and will
   * result in a BigInteger value of 0, whether signum is -1, 0 or 1.
   *
   * @param signum signum of the number (-1 for negative, 0 for zero, 1 for positive).
   * @param magnitude big-endian binary representation of the magnitude of the number.
   * @throws NumberFormatException {@code signum} is not one of the three legal values (-1, 0, and
   * 1), or {@code signum} is 0 and {@code magnitude} contains one or more non-zero bytes.
   */
  public BigInteger(int signum, byte[] magnitude) {
    this.mag = stripLeadingZeroBytes(magnitude);

    if (signum < -1 || signum > 1) {
      throw (new NumberFormatException("Invalid signum value"));
    }

    if (this.mag.length == 0) {
      this.signum = 0;
    } else {
      if (signum == 0) {
        throw (new NumberFormatException("signum-magnitude mismatch"));
      }
      this.signum = signum;
    }
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  /**
   * A constructor for internal use that translates the sign-magnitude
   * representation of a BigInteger into a BigInteger. It checks the
   * arguments and copies the magnitude so this constructor would be
   * safe for external use.
   */
  private BigInteger(int signum, int[] magnitude) {
    this.mag = stripLeadingZeroInts(magnitude);

    if (signum < -1 || signum > 1) {
      throw (new NumberFormatException("Invalid signum value"));
    }

    if (this.mag.length == 0) {
      this.signum = 0;
    } else {
      if (signum == 0) {
        throw (new NumberFormatException("signum-magnitude mismatch"));
      }
      this.signum = signum;
    }
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  /**
   * Translates the String representation of a BigInteger in the
   * specified radix into a BigInteger.  The String representation
   * consists of an optional minus or plus sign followed by a
   * sequence of one or more digits in the specified radix.  The
   * character-to-digit mapping is provided by {@code
   * Character.digit}.  The String may not contain any extraneous
   * characters (whitespace, for example).
   *
   * @param val String representation of BigInteger.
   * @param radix radix to be used in interpreting {@code val}.
   * @throws NumberFormatException {@code val} is not a valid representation of a BigInteger in the
   * specified radix, or {@code radix} is outside the range from {@link Character#MIN_RADIX} to
   * {@link Character#MAX_RADIX}, inclusive.
   * @see Character#digit
   */
  public BigInteger(String val, int radix) {
    int cursor = 0, numDigits;
    final int len = val.length();

    if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) {
      throw new NumberFormatException("Radix out of range");
    }
    if (len == 0) {
      throw new NumberFormatException("Zero length BigInteger");
    }

    // Check for at most one leading sign
    int sign = 1;
    int index1 = val.lastIndexOf('-');
    int index2 = val.lastIndexOf('+');
    if (index1 >= 0) {
      if (index1 != 0 || index2 >= 0) {
        throw new NumberFormatException("Illegal embedded sign character");
      }
      sign = -1;
      cursor = 1;
    } else if (index2 >= 0) {
      if (index2 != 0) {
        throw new NumberFormatException("Illegal embedded sign character");
      }
      cursor = 1;
    }
    if (cursor == len) {
      throw new NumberFormatException("Zero length BigInteger");
    }

    // Skip leading zeros and compute number of digits in magnitude
    while (cursor < len &&
        Character.digit(val.charAt(cursor), radix) == 0) {
      cursor++;
    }

    if (cursor == len) {
      signum = 0;
      mag = ZERO.mag;
      return;
    }

    numDigits = len - cursor;
    signum = sign;

    // Pre-allocate array of expected size. May be too large but can
    // never be too small. Typically exact.
    long numBits = ((numDigits * bitsPerDigit[radix]) >>> 10) + 1;
    if (numBits + 31 >= (1L << 32)) {
      reportOverflow();
    }
    int numWords = (int) (numBits + 31) >>> 5;
    int[] magnitude = new int[numWords];

    // Process first (potentially short) digit group
    int firstGroupLen = numDigits % digitsPerInt[radix];
    if (firstGroupLen == 0) {
      firstGroupLen = digitsPerInt[radix];
    }
    String group = val.substring(cursor, cursor += firstGroupLen);
    magnitude[numWords - 1] = Integer.parseInt(group, radix);
    if (magnitude[numWords - 1] < 0) {
      throw new NumberFormatException("Illegal digit");
    }

    // Process remaining digit groups
    int superRadix = intRadix[radix];
    int groupVal = 0;
    while (cursor < len) {
      group = val.substring(cursor, cursor += digitsPerInt[radix]);
      groupVal = Integer.parseInt(group, radix);
      if (groupVal < 0) {
        throw new NumberFormatException("Illegal digit");
      }
      destructiveMulAdd(magnitude, superRadix, groupVal);
    }
    // Required for cases where the array was overallocated.
    mag = trustedStripLeadingZeroInts(magnitude);
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  /*
     * Constructs a new BigInteger using a char array with radix=10.
     * Sign is precalculated outside and not allowed in the val.
     */
  BigInteger(char[] val, int sign, int len) {
    int cursor = 0, numDigits;

    // Skip leading zeros and compute number of digits in magnitude
    while (cursor < len && Character.digit(val[cursor], 10) == 0) {
      cursor++;
    }
    if (cursor == len) {
      signum = 0;
      mag = ZERO.mag;
      return;
    }

    numDigits = len - cursor;
    signum = sign;
    // Pre-allocate array of expected size
    int numWords;
    if (len < 10) {
      numWords = 1;
    } else {
      long numBits = ((numDigits * bitsPerDigit[10]) >>> 10) + 1;
      if (numBits + 31 >= (1L << 32)) {
        reportOverflow();
      }
      numWords = (int) (numBits + 31) >>> 5;
    }
    int[] magnitude = new int[numWords];

    // Process first (potentially short) digit group
    int firstGroupLen = numDigits % digitsPerInt[10];
    if (firstGroupLen == 0) {
      firstGroupLen = digitsPerInt[10];
    }
    magnitude[numWords - 1] = parseInt(val, cursor, cursor += firstGroupLen);

    // Process remaining digit groups
    while (cursor < len) {
      int groupVal = parseInt(val, cursor, cursor += digitsPerInt[10]);
      destructiveMulAdd(magnitude, intRadix[10], groupVal);
    }
    mag = trustedStripLeadingZeroInts(magnitude);
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  // Create an integer with the digits between the two indexes
  // Assumes start < end. The result may be negative, but it
  // is to be treated as an unsigned value.
  private int parseInt(char[] source, int start, int end) {
    int result = Character.digit(source[start++], 10);
    if (result == -1) {
      throw new NumberFormatException(new String(source));
    }

    for (int index = start; index < end; index++) {
      int nextVal = Character.digit(source[index], 10);
      if (nextVal == -1) {
        throw new NumberFormatException(new String(source));
      }
      result = 10 * result + nextVal;
    }

    return result;
  }

  // bitsPerDigit in the given radix times 1024
  // Rounded up to avoid underallocation.
  private static long bitsPerDigit[] = {0, 0,
      1024, 1624, 2048, 2378, 2648, 2875, 3072, 3247, 3402, 3543, 3672,
      3790, 3899, 4001, 4096, 4186, 4271, 4350, 4426, 4498, 4567, 4633,
      4696, 4756, 4814, 4870, 4923, 4975, 5025, 5074, 5120, 5166, 5210,
      5253, 5295};

  // Multiply x array times word y in place, and add word z
  private static void destructiveMulAdd(int[] x, int y, int z) {
    // Perform the multiplication word by word
    long ylong = y & LONG_MASK;
    long zlong = z & LONG_MASK;
    int len = x.length;

    long product = 0;
    long carry = 0;
    for (int i = len - 1; i >= 0; i--) {
      product = ylong * (x[i] & LONG_MASK) + carry;
      x[i] = (int) product;
      carry = product >>> 32;
    }

    // Perform the addition
    long sum = (x[len - 1] & LONG_MASK) + zlong;
    x[len - 1] = (int) sum;
    carry = sum >>> 32;
    for (int i = len - 2; i >= 0; i--) {
      sum = (x[i] & LONG_MASK) + carry;
      x[i] = (int) sum;
      carry = sum >>> 32;
    }
  }

  /**
   * Translates the decimal String representation of a BigInteger into a
   * BigInteger.  The String representation consists of an optional minus
   * sign followed by a sequence of one or more decimal digits.  The
   * character-to-digit mapping is provided by {@code Character.digit}.
   * The String may not contain any extraneous characters (whitespace, for
   * example).
   *
   * @param val decimal String representation of BigInteger.
   * @throws NumberFormatException {@code val} is not a valid representation of a BigInteger.
   * @see Character#digit
   */
  public BigInteger(String val) {
    this(val, 10);
  }

  /**
   * Constructs a randomly generated BigInteger, uniformly distributed over
   * the range 0 to (2<sup>{@code numBits}</sup> - 1), inclusive.
   * The uniformity of the distribution assumes that a fair source of random
   * bits is provided in {@code rnd}.  Note that this constructor always
   * constructs a non-negative BigInteger.
   *
   * @param numBits maximum bitLength of the new BigInteger.
   * @param rnd source of randomness to be used in computing the new BigInteger.
   * @throws IllegalArgumentException {@code numBits} is negative.
   * @see #bitLength()
   */
  public BigInteger(int numBits, Random rnd) {
    this(1, randomBits(numBits, rnd));
  }

  private static byte[] randomBits(int numBits, Random rnd) {
    if (numBits < 0) {
      throw new IllegalArgumentException("numBits must be non-negative");
    }
    int numBytes = (int) (((long) numBits + 7) / 8); // avoid overflow
    byte[] randomBits = new byte[numBytes];

    // Generate random bytes and mask out any excess bits
    if (numBytes > 0) {
      rnd.nextBytes(randomBits);
      int excessBits = 8 * numBytes - numBits;
      randomBits[0] &= (1 << (8 - excessBits)) - 1;
    }
    return randomBits;
  }

  /**
   * Constructs a randomly generated positive BigInteger that is probably
   * prime, with the specified bitLength.
   *
   * <p>It is recommended that the {@link #probablePrime probablePrime}
   * method be used in preference to this constructor unless there
   * is a compelling need to specify a certainty.
   *
   * @param bitLength bitLength of the returned BigInteger.
   * @param certainty a measure of the uncertainty that the caller is willing to tolerate.  The
   * probability that the new BigInteger represents a prime number will exceed (1 - 1/2<sup>{@code
   * certainty}</sup>).  The execution time of this constructor is proportional to the value of this
   * parameter.
   * @param rnd source of random bits used to select candidates to be tested for primality.
   * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
   * @see #bitLength()
   */
  public BigInteger(int bitLength, int certainty, Random rnd) {
    BigInteger prime;

    if (bitLength < 2) {
      throw new ArithmeticException("bitLength < 2");
    }
    prime = (bitLength < SMALL_PRIME_THRESHOLD
        ? smallPrime(bitLength, certainty, rnd)
        : largePrime(bitLength, certainty, rnd));
    signum = 1;
    mag = prime.mag;
  }

  // Minimum size in bits that the requested prime number has
  // before we use the large prime number generating algorithms.
  // The cutoff of 95 was chosen empirically for best performance.
  private static final int SMALL_PRIME_THRESHOLD = 95;

  // Certainty required to meet the spec of probablePrime
  private static final int DEFAULT_PRIME_CERTAINTY = 100;

  /**
   * Returns a positive BigInteger that is probably prime, with the
   * specified bitLength. The probability that a BigInteger returned
   * by this method is composite does not exceed 2<sup>-100</sup>.
   *
   * @param bitLength bitLength of the returned BigInteger.
   * @param rnd source of random bits used to select candidates to be tested for primality.
   * @return a BigInteger of {@code bitLength} bits that is probably prime
   * @throws ArithmeticException {@code bitLength < 2} or {@code bitLength} is too large.
   * @see #bitLength()
   * @since 1.4
   */
  public static BigInteger probablePrime(int bitLength, Random rnd) {
    if (bitLength < 2) {
      throw new ArithmeticException("bitLength < 2");
    }

    return (bitLength < SMALL_PRIME_THRESHOLD ?
        smallPrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd) :
        largePrime(bitLength, DEFAULT_PRIME_CERTAINTY, rnd));
  }

  /**
   * Find a random number of the specified bitLength that is probably prime.
   * This method is used for smaller primes, its performance degrades on
   * larger bitlengths.
   *
   * This method assumes bitLength > 1.
   */
  private static BigInteger smallPrime(int bitLength, int certainty, Random rnd) {
    int magLen = (bitLength + 31) >>> 5;
    int temp[] = new int[magLen];
    int highBit = 1 << ((bitLength + 31) & 0x1f);  // High bit of high int
    int highMask = (highBit << 1) - 1;  // Bits to keep in high int

    while (true) {
      // Construct a candidate
      for (int i = 0; i < magLen; i++) {
        temp[i] = rnd.nextInt();
      }
      temp[0] = (temp[0] & highMask) | highBit;  // Ensure exact length
      if (bitLength > 2) {
        temp[magLen - 1] |= 1;  // Make odd if bitlen > 2
      }

      BigInteger p = new BigInteger(temp, 1);

      // Do cheap "pre-test" if applicable
      if (bitLength > 6) {
        long r = p.remainder(SMALL_PRIME_PRODUCT).longValue();
        if ((r % 3 == 0) || (r % 5 == 0) || (r % 7 == 0) || (r % 11 == 0) ||
            (r % 13 == 0) || (r % 17 == 0) || (r % 19 == 0) || (r % 23 == 0) ||
            (r % 29 == 0) || (r % 31 == 0) || (r % 37 == 0) || (r % 41 == 0)) {
          continue; // Candidate is composite; try another
        }
      }

      // All candidates of bitLength 2 and 3 are prime by this point
      if (bitLength < 4) {
        return p;
      }

      // Do expensive test if we survive pre-test (or it's inapplicable)
      if (p.primeToCertainty(certainty, rnd)) {
        return p;
      }
    }
  }

  private static final BigInteger SMALL_PRIME_PRODUCT
      = valueOf(3L * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41);

  /**
   * Find a random number of the specified bitLength that is probably prime.
   * This method is more appropriate for larger bitlengths since it uses
   * a sieve to eliminate most composites before using a more expensive
   * test.
   */
  private static BigInteger largePrime(int bitLength, int certainty, Random rnd) {
    BigInteger p;
    p = new BigInteger(bitLength, rnd).setBit(bitLength - 1);
    p.mag[p.mag.length - 1] &= 0xfffffffe;

    // Use a sieve length likely to contain the next prime number
    int searchLen = getPrimeSearchLen(bitLength);
    BitSieve searchSieve = new BitSieve(p, searchLen);
    BigInteger candidate = searchSieve.retrieve(p, certainty, rnd);

    while ((candidate == null) || (candidate.bitLength() != bitLength)) {
      p = p.add(BigInteger.valueOf(2 * searchLen));
      if (p.bitLength() != bitLength) {
        p = new BigInteger(bitLength, rnd).setBit(bitLength - 1);
      }
      p.mag[p.mag.length - 1] &= 0xfffffffe;
      searchSieve = new BitSieve(p, searchLen);
      candidate = searchSieve.retrieve(p, certainty, rnd);
    }
    return candidate;
  }

  /**
   * Returns the first integer greater than this {@code BigInteger} that
   * is probably prime.  The probability that the number returned by this
   * method is composite does not exceed 2<sup>-100</sup>. This method will
   * never skip over a prime when searching: if it returns {@code p}, there
   * is no prime {@code q} such that {@code this < q < p}.
   *
   * @return the first integer greater than this {@code BigInteger} that is probably prime.
   * @throws ArithmeticException {@code this < 0} or {@code this} is too large.
   * @since 1.5
   */
  public BigInteger nextProbablePrime() {
    if (this.signum < 0) {
      throw new ArithmeticException("start < 0: " + this);
    }

    // Handle trivial cases
    if ((this.signum == 0) || this.equals(ONE)) {
      return TWO;
    }

    BigInteger result = this.add(ONE);

    // Fastpath for small numbers
    if (result.bitLength() < SMALL_PRIME_THRESHOLD) {

      // Ensure an odd number
      if (!result.testBit(0)) {
        result = result.add(ONE);
      }

      while (true) {
        // Do cheap "pre-test" if applicable
        if (result.bitLength() > 6) {
          long r = result.remainder(SMALL_PRIME_PRODUCT).longValue();
          if ((r % 3 == 0) || (r % 5 == 0) || (r % 7 == 0) || (r % 11 == 0) ||
              (r % 13 == 0) || (r % 17 == 0) || (r % 19 == 0) || (r % 23 == 0) ||
              (r % 29 == 0) || (r % 31 == 0) || (r % 37 == 0) || (r % 41 == 0)) {
            result = result.add(TWO);
            continue; // Candidate is composite; try another
          }
        }

        // All candidates of bitLength 2 and 3 are prime by this point
        if (result.bitLength() < 4) {
          return result;
        }

        // The expensive test
        if (result.primeToCertainty(DEFAULT_PRIME_CERTAINTY, null)) {
          return result;
        }

        result = result.add(TWO);
      }
    }

    // Start at previous even number
    if (result.testBit(0)) {
      result = result.subtract(ONE);
    }

    // Looking for the next large prime
    int searchLen = getPrimeSearchLen(result.bitLength());

    while (true) {
      BitSieve searchSieve = new BitSieve(result, searchLen);
      BigInteger candidate = searchSieve.retrieve(result,
          DEFAULT_PRIME_CERTAINTY, null);
      if (candidate != null) {
        return candidate;
      }
      result = result.add(BigInteger.valueOf(2 * searchLen));
    }
  }

  private static int getPrimeSearchLen(int bitLength) {
    if (bitLength > PRIME_SEARCH_BIT_LENGTH_LIMIT + 1) {
      throw new ArithmeticException("Prime search implementation restriction on bitLength");
    }
    return bitLength / 20 * 64;
  }

  /**
   * Returns {@code true} if this BigInteger is probably prime,
   * {@code false} if it's definitely composite.
   *
   * This method assumes bitLength > 2.
   *
   * @param certainty a measure of the uncertainty that the caller is willing to tolerate: if the
   * call returns {@code true} the probability that this BigInteger is prime exceeds {@code (1 -
   * 1/2<sup>certainty</sup>)}.  The execution time of this method is proportional to the value of
   * this parameter.
   * @return {@code true} if this BigInteger is probably prime, {@code false} if it's definitely
   * composite.
   */
  boolean primeToCertainty(int certainty, Random random) {
    int rounds = 0;
    int n = (Math.min(certainty, Integer.MAX_VALUE - 1) + 1) / 2;

    // The relationship between the certainty and the number of rounds
    // we perform is given in the draft standard ANSI X9.80, "PRIME
    // NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
    int sizeInBits = this.bitLength();
    if (sizeInBits < 100) {
      rounds = 50;
      rounds = n < rounds ? n : rounds;
      return passesMillerRabin(rounds, random);
    }

    if (sizeInBits < 256) {
      rounds = 27;
    } else if (sizeInBits < 512) {
      rounds = 15;
    } else if (sizeInBits < 768) {
      rounds = 8;
    } else if (sizeInBits < 1024) {
      rounds = 4;
    } else {
      rounds = 2;
    }
    rounds = n < rounds ? n : rounds;

    return passesMillerRabin(rounds, random) && passesLucasLehmer();
  }

  /**
   * Returns true iff this BigInteger is a Lucas-Lehmer probable prime.
   *
   * The following assumptions are made:
   * This BigInteger is a positive, odd number.
   */
  private boolean passesLucasLehmer() {
    BigInteger thisPlusOne = this.add(ONE);

    // Step 1
    int d = 5;
    while (jacobiSymbol(d, this) != -1) {
      // 5, -7, 9, -11, ...
      d = (d < 0) ? Math.abs(d) + 2 : -(d + 2);
    }

    // Step 2
    BigInteger u = lucasLehmerSequence(d, thisPlusOne, this);

    // Step 3
    return u.mod(this).equals(ZERO);
  }

  /**
   * Computes Jacobi(p,n).
   * Assumes n positive, odd, n>=3.
   */
  private static int jacobiSymbol(int p, BigInteger n) {
    if (p == 0) {
      return 0;
    }

    // Algorithm and comments adapted from Colin Plumb's C library.
    int j = 1;
    int u = n.mag[n.mag.length - 1];

    // Make p positive
    if (p < 0) {
      p = -p;
      int n8 = u & 7;
      if ((n8 == 3) || (n8 == 7)) {
        j = -j; // 3 (011) or 7 (111) mod 8
      }
    }

    // Get rid of factors of 2 in p
    while ((p & 3) == 0) {
      p >>= 2;
    }
    if ((p & 1) == 0) {
      p >>= 1;
      if (((u ^ (u >> 1)) & 2) != 0) {
        j = -j; // 3 (011) or 5 (101) mod 8
      }
    }
    if (p == 1) {
      return j;
    }
    // Then, apply quadratic reciprocity
    if ((p & u & 2) != 0)   // p = u = 3 (mod 4)?
    {
      j = -j;
    }
    // And reduce u mod p
    u = n.mod(BigInteger.valueOf(p)).intValue();

    // Now compute Jacobi(u,p), u < p
    while (u != 0) {
      while ((u & 3) == 0) {
        u >>= 2;
      }
      if ((u & 1) == 0) {
        u >>= 1;
        if (((p ^ (p >> 1)) & 2) != 0) {
          j = -j;     // 3 (011) or 5 (101) mod 8
        }
      }
      if (u == 1) {
        return j;
      }
      // Now both u and p are odd, so use quadratic reciprocity
      assert (u < p);
      int t = u;
      u = p;
      p = t;
      if ((u & p & 2) != 0) // u = p = 3 (mod 4)?
      {
        j = -j;
      }
      // Now u >= p, so it can be reduced
      u %= p;
    }
    return 0;
  }

  private static BigInteger lucasLehmerSequence(int z, BigInteger k, BigInteger n) {
    BigInteger d = BigInteger.valueOf(z);
    BigInteger u = ONE;
    BigInteger u2;
    BigInteger v = ONE;
    BigInteger v2;

    for (int i = k.bitLength() - 2; i >= 0; i--) {
      u2 = u.multiply(v).mod(n);

      v2 = v.square().add(d.multiply(u.square())).mod(n);
      if (v2.testBit(0)) {
        v2 = v2.subtract(n);
      }

      v2 = v2.shiftRight(1);

      u = u2;
      v = v2;
      if (k.testBit(i)) {
        u2 = u.add(v).mod(n);
        if (u2.testBit(0)) {
          u2 = u2.subtract(n);
        }

        u2 = u2.shiftRight(1);
        v2 = v.add(d.multiply(u)).mod(n);
        if (v2.testBit(0)) {
          v2 = v2.subtract(n);
        }
        v2 = v2.shiftRight(1);

        u = u2;
        v = v2;
      }
    }
    return u;
  }

  /**
   * Returns true iff this BigInteger passes the specified number of
   * Miller-Rabin tests. This test is taken from the DSA spec (NIST FIPS
   * 186-2).
   *
   * The following assumptions are made:
   * This BigInteger is a positive, odd number greater than 2.
   * iterations<=50.
   */
  private boolean passesMillerRabin(int iterations, Random rnd) {
    // Find a and m such that m is odd and this == 1 + 2**a * m
    BigInteger thisMinusOne = this.subtract(ONE);
    BigInteger m = thisMinusOne;
    int a = m.getLowestSetBit();
    m = m.shiftRight(a);

    // Do the tests
    if (rnd == null) {
      rnd = ThreadLocalRandom.current();
    }
    for (int i = 0; i < iterations; i++) {
      // Generate a uniform random on (1, this)
      BigInteger b;
      do {
        b = new BigInteger(this.bitLength(), rnd);
      } while (b.compareTo(ONE) <= 0 || b.compareTo(this) >= 0);

      int j = 0;
      BigInteger z = b.modPow(m, this);
      while (!((j == 0 && z.equals(ONE)) || z.equals(thisMinusOne))) {
        if (j > 0 && z.equals(ONE) || ++j == a) {
          return false;
        }
        z = z.modPow(TWO, this);
      }
    }
    return true;
  }

  /**
   * This internal constructor differs from its public cousin
   * with the arguments reversed in two ways: it assumes that its
   * arguments are correct, and it doesn't copy the magnitude array.
   */
  BigInteger(int[] magnitude, int signum) {
    this.signum = (magnitude.length == 0 ? 0 : signum);
    this.mag = magnitude;
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  /**
   * This private constructor is for internal use and assumes that its
   * arguments are correct.
   */
  private BigInteger(byte[] magnitude, int signum) {
    this.signum = (magnitude.length == 0 ? 0 : signum);
    this.mag = stripLeadingZeroBytes(magnitude);
    if (mag.length >= MAX_MAG_LENGTH) {
      checkRange();
    }
  }

  /**
   * Throws an {@code ArithmeticException} if the {@code BigInteger} would be
   * out of the supported range.
   *
   * @throws ArithmeticException if {@code this} exceeds the supported range.
   */
  private void checkRange() {
    if (mag.length > MAX_MAG_LENGTH || mag.length == MAX_MAG_LENGTH && mag[0] < 0) {
      reportOverflow();
    }
  }

  private static void reportOverflow() {
    throw new ArithmeticException("BigInteger would overflow supported range");
  }

  //Static Factory Methods

  /**
   * Returns a BigInteger whose value is equal to that of the
   * specified {@code long}.  This "static factory method" is
   * provided in preference to a ({@code long}) constructor
   * because it allows for reuse of frequently used BigIntegers.
   *
   * @param val value of the BigInteger to return.
   * @return a BigInteger with the specified value.
   */
  public static BigInteger valueOf(long val) {
    // If -MAX_CONSTANT < val < MAX_CONSTANT, return stashed constant
    if (val == 0) {
      return ZERO;
    }
    if (val > 0 && val <= MAX_CONSTANT) {
      return posConst[(int) val];
    } else if (val < 0 && val >= -MAX_CONSTANT) {
      return negConst[(int) -val];
    }

    return new BigInteger(val);
  }

  /**
   * Constructs a BigInteger with the specified value, which may not be zero.
   */
  private BigInteger(long val) {
    if (val < 0) {
      val = -val;
      signum = -1;
    } else {
      signum = 1;
    }

    int highWord = (int) (val >>> 32);
    if (highWord == 0) {
      mag = new int[1];
      mag[0] = (int) val;
    } else {
      mag = new int[2];
      mag[0] = highWord;
      mag[1] = (int) val;
    }
  }

  /**
   * Returns a BigInteger with the given two's complement representation.
   * Assumes that the input array will not be modified (the returned
   * BigInteger will reference the input array if feasible).
   */
  private static BigInteger valueOf(int val[]) {
    return (val[0] > 0 ? new BigInteger(val, 1) : new BigInteger(val));
  }

  // Constants

  /**
   * Initialize static constant array when class is loaded.
   */
  private final static int MAX_CONSTANT = 16;
  private static BigInteger posConst[] = new BigInteger[MAX_CONSTANT + 1];
  private static BigInteger negConst[] = new BigInteger[MAX_CONSTANT + 1];

  /**
   * The cache of powers of each radix.  This allows us to not have to
   * recalculate powers of radix^(2^n) more than once.  This speeds
   * Schoenhage recursive base conversion significantly.
   */
  private static volatile BigInteger[][] powerCache;

  /**
   * The cache of logarithms of radices for base conversion.
   */
  private static final double[] logCache;

  /**
   * The natural log of 2.  This is used in computing cache indices.
   */
  private static final double LOG_TWO = Math.log(2.0);

  static {
    for (int i = 1; i <= MAX_CONSTANT; i++) {
      int[] magnitude = new int[1];
      magnitude[0] = i;
      posConst[i] = new BigInteger(magnitude, 1);
      negConst[i] = new BigInteger(magnitude, -1);
    }

        /*
         * Initialize the cache of radix^(2^x) values used for base conversion
         * with just the very first value.  Additional values will be created
         * on demand.
         */
    powerCache = new BigInteger[Character.MAX_RADIX + 1][];
    logCache = new double[Character.MAX_RADIX + 1];

    for (int i = Character.MIN_RADIX; i <= Character.MAX_RADIX; i++) {
      powerCache[i] = new BigInteger[]{BigInteger.valueOf(i)};
      logCache[i] = Math.log(i);
    }
  }

  /**
   * The BigInteger constant zero.
   *
   * @since 1.2
   */
  public static final BigInteger ZERO = new BigInteger(new int[0], 0);

  /**
   * The BigInteger constant one.
   *
   * @since 1.2
   */
  public static final BigInteger ONE = valueOf(1);

  /**
   * The BigInteger constant two.  (Not exported.)
   */
  private static final BigInteger TWO = valueOf(2);

  /**
   * The BigInteger constant -1.  (Not exported.)
   */
  private static final BigInteger NEGATIVE_ONE = valueOf(-1);

  /**
   * The BigInteger constant ten.
   *
   * @since 1.5
   */
  public static final BigInteger TEN = valueOf(10);

  // Arithmetic Operations

  /**
   * Returns a BigInteger whose value is {@code (this + val)}.
   *
   * @param val value to be added to this BigInteger.
   * @return {@code this + val}
   */
  public BigInteger add(BigInteger val) {
    if (val.signum == 0) {
      return this;
    }
    if (signum == 0) {
      return val;
    }
    if (val.signum == signum) {
      return new BigInteger(add(mag, val.mag), signum);
    }

    int cmp = compareMagnitude(val);
    if (cmp == 0) {
      return ZERO;
    }
    int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
        : subtract(val.mag, mag));
    resultMag = trustedStripLeadingZeroInts(resultMag);

    return new BigInteger(resultMag, cmp == signum ? 1 : -1);
  }

  /**
   * Package private methods used by BigDecimal code to add a BigInteger
   * with a long. Assumes val is not equal to INFLATED.
   */
  BigInteger add(long val) {
    if (val == 0) {
      return this;
    }
    if (signum == 0) {
      return valueOf(val);
    }
    if (Long.signum(val) == signum) {
      return new BigInteger(add(mag, Math.abs(val)), signum);
    }
    int cmp = compareMagnitude(val);
    if (cmp == 0) {
      return ZERO;
    }
    int[] resultMag = (cmp > 0 ? subtract(mag, Math.abs(val)) : subtract(Math.abs(val), mag));
    resultMag = trustedStripLeadingZeroInts(resultMag);
    return new BigInteger(resultMag, cmp == signum ? 1 : -1);
  }

  /**
   * Adds the contents of the int array x and long value val. This
   * method allocates a new int array to hold the answer and returns
   * a reference to that array.  Assumes x.length &gt; 0 and val is
   * non-negative
   */
  private static int[] add(int[] x, long val) {
    int[] y;
    long sum = 0;
    int xIndex = x.length;
    int[] result;
    int highWord = (int) (val >>> 32);
    if (highWord == 0) {
      result = new int[xIndex];
      sum = (x[--xIndex] & LONG_MASK) + val;
      result[xIndex] = (int) sum;
    } else {
      if (xIndex == 1) {
        result = new int[2];
        sum = val + (x[0] & LONG_MASK);
        result[1] = (int) sum;
        result[0] = (int) (sum >>> 32);
        return result;
      } else {
        result = new int[xIndex];
        sum = (x[--xIndex] & LONG_MASK) + (val & LONG_MASK);
        result[xIndex] = (int) sum;
        sum = (x[--xIndex] & LONG_MASK) + (highWord & LONG_MASK) + (sum >>> 32);
        result[xIndex] = (int) sum;
      }
    }
    // Copy remainder of longer number while carry propagation is required
    boolean carry = (sum >>> 32 != 0);
    while (xIndex > 0 && carry) {
      carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
    }
    // Copy remainder of longer number
    while (xIndex > 0) {
      result[--xIndex] = x[xIndex];
    }
    // Grow result if necessary
    if (carry) {
      int bigger[] = new int[result.length + 1];
      System.arraycopy(result, 0, bigger, 1, result.length);
      bigger[0] = 0x01;
      return bigger;
    }
    return result;
  }

  /**
   * Adds the contents of the int arrays x and y. This method allocates
   * a new int array to hold the answer and returns a reference to that
   * array.
   */
  private static int[] add(int[] x, int[] y) {
    // If x is shorter, swap the two arrays
    if (x.length < y.length) {
      int[] tmp = x;
      x = y;
      y = tmp;
    }

    int xIndex = x.length;
    int yIndex = y.length;
    int result[] = new int[xIndex];
    long sum = 0;
    if (yIndex == 1) {
      sum = (x[--xIndex] & LONG_MASK) + (y[0] & LONG_MASK);
      result[xIndex] = (int) sum;
    } else {
      // Add common parts of both numbers
      while (yIndex > 0) {
        sum = (x[--xIndex] & LONG_MASK) +
            (y[--yIndex] & LONG_MASK) + (sum >>> 32);
        result[xIndex] = (int) sum;
      }
    }
    // Copy remainder of longer number while carry propagation is required
    boolean carry = (sum >>> 32 != 0);
    while (xIndex > 0 && carry) {
      carry = ((result[--xIndex] = x[xIndex] + 1) == 0);
    }

    // Copy remainder of longer number
    while (xIndex > 0) {
      result[--xIndex] = x[xIndex];
    }

    // Grow result if necessary
    if (carry) {
      int bigger[] = new int[result.length + 1];
      System.arraycopy(result, 0, bigger, 1, result.length);
      bigger[0] = 0x01;
      return bigger;
    }
    return result;
  }

  private static int[] subtract(long val, int[] little) {
    int highWord = (int) (val >>> 32);
    if (highWord == 0) {
      int result[] = new int[1];
      result[0] = (int) (val - (little[0] & LONG_MASK));
      return result;
    } else {
      int result[] = new int[2];
      if (little.length == 1) {
        long difference = ((int) val & LONG_MASK) - (little[0] & LONG_MASK);
        result[1] = (int) difference;
        // Subtract remainder of longer number while borrow propagates
        boolean borrow = (difference >> 32 != 0);
        if (borrow) {
          result[0] = highWord - 1;
        } else {        // Copy remainder of longer number
          result[0] = highWord;
        }
        return result;
      } else { // little.length == 2
        long difference = ((int) val & LONG_MASK) - (little[1] & LONG_MASK);
        result[1] = (int) difference;
        difference = (highWord & LONG_MASK) - (little[0] & LONG_MASK) + (difference >> 32);
        result[0] = (int) difference;
        return result;
      }
    }
  }

  /**
   * Subtracts the contents of the second argument (val) from the
   * first (big).  The first int array (big) must represent a larger number
   * than the second.  This method allocates the space necessary to hold the
   * answer.
   * assumes val &gt;= 0
   */
  private static int[] subtract(int[] big, long val) {
    int highWord = (int) (val >>> 32);
    int bigIndex = big.length;
    int result[] = new int[bigIndex];
    long difference = 0;

    if (highWord == 0) {
      difference = (big[--bigIndex] & LONG_MASK) - val;
      result[bigIndex] = (int) difference;
    } else {
      difference = (big[--bigIndex] & LONG_MASK) - (val & LONG_MASK);
      result[bigIndex] = (int) difference;
      difference = (big[--bigIndex] & LONG_MASK) - (highWord & LONG_MASK) + (difference >> 32);
      result[bigIndex] = (int) difference;
    }

    // Subtract remainder of longer number while borrow propagates
    boolean borrow = (difference >> 32 != 0);
    while (bigIndex > 0 && borrow) {
      borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
    }

    // Copy remainder of longer number
    while (bigIndex > 0) {
      result[--bigIndex] = big[bigIndex];
    }

    return result;
  }

  /**
   * Returns a BigInteger whose value is {@code (this - val)}.
   *
   * @param val value to be subtracted from this BigInteger.
   * @return {@code this - val}
   */
  public BigInteger subtract(BigInteger val) {
    if (val.signum == 0) {
      return this;
    }
    if (signum == 0) {
      return val.negate();
    }
    if (val.signum != signum) {
      return new BigInteger(add(mag, val.mag), signum);
    }

    int cmp = compareMagnitude(val);
    if (cmp == 0) {
      return ZERO;
    }
    int[] resultMag = (cmp > 0 ? subtract(mag, val.mag)
        : subtract(val.mag, mag));
    resultMag = trustedStripLeadingZeroInts(resultMag);
    return new BigInteger(resultMag, cmp == signum ? 1 : -1);
  }

  /**
   * Subtracts the contents of the second int arrays (little) from the
   * first (big).  The first int array (big) must represent a larger number
   * than the second.  This method allocates the space necessary to hold the
   * answer.
   */
  private static int[] subtract(int[] big, int[] little) {
    int bigIndex = big.length;
    int result[] = new int[bigIndex];
    int littleIndex = little.length;
    long difference = 0;

    // Subtract common parts of both numbers
    while (littleIndex > 0) {
      difference = (big[--bigIndex] & LONG_MASK) -
          (little[--littleIndex] & LONG_MASK) +
          (difference >> 32);
      result[bigIndex] = (int) difference;
    }

    // Subtract remainder of longer number while borrow propagates
    boolean borrow = (difference >> 32 != 0);
    while (bigIndex > 0 && borrow) {
      borrow = ((result[--bigIndex] = big[bigIndex] - 1) == -1);
    }

    // Copy remainder of longer number
    while (bigIndex > 0) {
      result[--bigIndex] = big[bigIndex];
    }

    return result;
  }

  /**
   * Returns a BigInteger whose value is {@code (this * val)}.
   *
   * @param val value to be multiplied by this BigInteger.
   * @return {@code this * val}
   * @implNote An implementation may offer better algorithmic performance when {@code val == this}.
   */
  public BigInteger multiply(BigInteger val) {
    if (val.signum == 0 || signum == 0) {
      return ZERO;
    }

    int xlen = mag.length;

    if (val == this && xlen > MULTIPLY_SQUARE_THRESHOLD) {
      return square();
    }

    int ylen = val.mag.length;

    if ((xlen < KARATSUBA_THRESHOLD) || (ylen < KARATSUBA_THRESHOLD)) {
      int resultSign = signum == val.signum ? 1 : -1;
      if (val.mag.length == 1) {
        return multiplyByInt(mag, val.mag[0], resultSign);
      }
      if (mag.length == 1) {
        return multiplyByInt(val.mag, mag[0], resultSign);
      }
      int[] result = multiplyToLen(mag, xlen,
          val.mag, ylen, null);
      result = trustedStripLeadingZeroInts(result);
      return new BigInteger(result, resultSign);
    } else {
      if ((xlen < TOOM_COOK_THRESHOLD) && (ylen < TOOM_COOK_THRESHOLD)) {
        return multiplyKaratsuba(this, val);
      } else {
        return multiplyToomCook3(this, val);
      }
    }
  }

  private static BigInteger multiplyByInt(int[] x, int y, int sign) {
    if (Integer.bitCount(y) == 1) {
      return new BigInteger(shiftLeft(x, Integer.numberOfTrailingZeros(y)), sign);
    }
    int xlen = x.length;
    int[] rmag = new int[xlen + 1];
    long carry = 0;
    long yl = y & LONG_MASK;
    int rstart = rmag.length - 1;
    for (int i = xlen - 1; i >= 0; i--) {
      long product = (x[i] & LONG_MASK) * yl + carry;
      rmag[rstart--] = (int) product;
      carry = product >>> 32;
    }
    if (carry == 0L) {
      rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
    } else {
      rmag[rstart] = (int) carry;
    }
    return new BigInteger(rmag, sign);
  }

  /**
   * Package private methods used by BigDecimal code to multiply a BigInteger
   * with a long. Assumes v is not equal to INFLATED.
   */
  BigInteger multiply(long v) {
    if (v == 0 || signum == 0) {
      return ZERO;
    }
    if (v == BigDecimal.INFLATED) {
      return multiply(BigInteger.valueOf(v));
    }
    int rsign = (v > 0 ? signum : -signum);
    if (v < 0) {
      v = -v;
    }
    long dh = v >>> 32;      // higher order bits
    long dl = v & LONG_MASK; // lower order bits

    int xlen = mag.length;
    int[] value = mag;
    int[] rmag = (dh == 0L) ? (new int[xlen + 1]) : (new int[xlen + 2]);
    long carry = 0;
    int rstart = rmag.length - 1;
    for (int i = xlen - 1; i >= 0; i--) {
      long product = (value[i] & LONG_MASK) * dl + carry;
      rmag[rstart--] = (int) product;
      carry = product >>> 32;
    }
    rmag[rstart] = (int) carry;
    if (dh != 0L) {
      carry = 0;
      rstart = rmag.length - 2;
      for (int i = xlen - 1; i >= 0; i--) {
        long product = (value[i] & LONG_MASK) * dh +
            (rmag[rstart] & LONG_MASK) + carry;
        rmag[rstart--] = (int) product;
        carry = product >>> 32;
      }
      rmag[0] = (int) carry;
    }
    if (carry == 0L) {
      rmag = java.util.Arrays.copyOfRange(rmag, 1, rmag.length);
    }
    return new BigInteger(rmag, rsign);
  }

  /**
   * Multiplies int arrays x and y to the specified lengths and places
   * the result into z. There will be no leading zeros in the resultant array.
   */
  private int[] multiplyToLen(int[] x, int xlen, int[] y, int ylen, int[] z) {
    int xstart = xlen - 1;
    int ystart = ylen - 1;

    if (z == null || z.length < (xlen + ylen)) {
      z = new int[xlen + ylen];
    }

    long carry = 0;
    for (int j = ystart, k = ystart + 1 + xstart; j >= 0; j--, k--) {
      long product = (y[j] & LONG_MASK) *
          (x[xstart] & LONG_MASK) + carry;
      z[k] = (int) product;
      carry = product >>> 32;
    }
    z[xstart] = (int) carry;

    for (int i = xstart - 1; i >= 0; i--) {
      carry = 0;
      for (int j = ystart, k = ystart + 1 + i; j >= 0; j--, k--) {
        long product = (y[j] & LONG_MASK) *
            (x[i] & LONG_MASK) +
            (z[k] & LONG_MASK) + carry;
        z[k] = (int) product;
        carry = product >>> 32;
      }
      z[i] = (int) carry;
    }
    return z;
  }

  /**
   * Multiplies two BigIntegers using the Karatsuba multiplication
   * algorithm.  This is a recursive divide-and-conquer algorithm which is
   * more efficient for large numbers than what is commonly called the
   * "grade-school" algorithm used in multiplyToLen.  If the numbers to be
   * multiplied have length n, the "grade-school" algorithm has an
   * asymptotic complexity of O(n^2).  In contrast, the Karatsuba algorithm
   * has complexity of O(n^(log2(3))), or O(n^1.585).  It achieves this
   * increased performance by doing 3 multiplies instead of 4 when
   * evaluating the product.  As it has some overhead, should be used when
   * both numbers are larger than a certain threshold (found
   * experimentally).
   *
   * See:  http://en.wikipedia.org/wiki/Karatsuba_algorithm
   */
  private static BigInteger multiplyKaratsuba(BigInteger x, BigInteger y) {
    int xlen = x.mag.length;
    int ylen = y.mag.length;

    // The number of ints in each half of the number.
    int half = (Math.max(xlen, ylen) + 1) / 2;

    // xl and yl are the lower halves of x and y respectively,
    // xh and yh are the upper halves.
    BigInteger xl = x.getLower(half);
    BigInteger xh = x.getUpper(half);
    BigInteger yl = y.getLower(half);
    BigInteger yh = y.getUpper(half);

    BigInteger p1 = xh.multiply(yh);  // p1 = xh*yh
    BigInteger p2 = xl.multiply(yl);  // p2 = xl*yl

    // p3=(xh+xl)*(yh+yl)
    BigInteger p3 = xh.add(xl).multiply(yh.add(yl));

    // result = p1 * 2^(32*2*half) + (p3 - p1 - p2) * 2^(32*half) + p2
    BigInteger result = p1.shiftLeft(32 * half).add(p3.subtract(p1).subtract(p2))
        .shiftLeft(32 * half).add(p2);

    if (x.signum != y.signum) {
      return result.negate();
    } else {
      return result;
    }
  }

  /**
   * Multiplies two BigIntegers using a 3-way Toom-Cook multiplication
   * algorithm.  This is a recursive divide-and-conquer algorithm which is
   * more efficient for large numbers than what is commonly called the
   * "grade-school" algorithm used in multiplyToLen.  If the numbers to be
   * multiplied have length n, the "grade-school" algorithm has an
   * asymptotic complexity of O(n^2).  In contrast, 3-way Toom-Cook has a
   * complexity of about O(n^1.465).  It achieves this increased asymptotic
   * performance by breaking each number into three parts and by doing 5
   * multiplies instead of 9 when evaluating the product.  Due to overhead
   * (additions, shifts, and one division) in the Toom-Cook algorithm, it
   * should only be used when both numbers are larger than a certain
   * threshold (found experimentally).  This threshold is generally larger
   * than that for Karatsuba multiplication, so this algorithm is generally
   * only used when numbers become significantly larger.
   *
   * The algorithm used is the "optimal" 3-way Toom-Cook algorithm outlined
   * by Marco Bodrato.
   *
   * See: http://bodrato.it/toom-cook/
   * http://bodrato.it/papers/#WAIFI2007
   *
   * "Towards Optimal Toom-Cook Multiplication for Univariate and
   * Multivariate Polynomials in Characteristic 2 and 0." by Marco BODRATO;
   * In C.Carlet and B.Sunar, Eds., "WAIFI'07 proceedings", p. 116-133,
   * LNCS #4547. Springer, Madrid, Spain, June 21-22, 2007.
   */
  private static BigInteger multiplyToomCook3(BigInteger a, BigInteger b) {
    int alen = a.mag.length;
    int blen = b.mag.length;

    int largest = Math.max(alen, blen);

    // k is the size (in ints) of the lower-order slices.
    int k = (largest + 2) / 3;   // Equal to ceil(largest/3)

    // r is the size (in ints) of the highest-order slice.
    int r = largest - 2 * k;

    // Obtain slices of the numbers. a2 and b2 are the most significant
    // bits of the numbers a and b, and a0 and b0 the least significant.
    BigInteger a0, a1, a2, b0, b1, b2;
    a2 = a.getToomSlice(k, r, 0, largest);
    a1 = a.getToomSlice(k, r, 1, largest);
    a0 = a.getToomSlice(k, r, 2, largest);
    b2 = b.getToomSlice(k, r, 0, largest);
    b1 = b.getToomSlice(k, r, 1, largest);
    b0 = b.getToomSlice(k, r, 2, largest);

    BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1, db1;

    v0 = a0.multiply(b0);
    da1 = a2.add(a0);
    db1 = b2.add(b0);
    vm1 = da1.subtract(a1).multiply(db1.subtract(b1));
    da1 = da1.add(a1);
    db1 = db1.add(b1);
    v1 = da1.multiply(db1);
    v2 = da1.add(a2).shiftLeft(1).subtract(a0).multiply(
        db1.add(b2).shiftLeft(1).subtract(b0));
    vinf = a2.multiply(b2);

    // The algorithm requires two divisions by 2 and one by 3.
    // All divisions are known to be exact, that is, they do not produce
    // remainders, and all results are positive.  The divisions by 2 are
    // implemented as right shifts which are relatively efficient, leaving
    // only an exact division by 3, which is done by a specialized
    // linear-time algorithm.
    t2 = v2.subtract(vm1).exactDivideBy3();
    tm1 = v1.subtract(vm1).shiftRight(1);
    t1 = v1.subtract(v0);
    t2 = t2.subtract(t1).shiftRight(1);
    t1 = t1.subtract(tm1).subtract(vinf);
    t2 = t2.subtract(vinf.shiftLeft(1));
    tm1 = tm1.subtract(t2);

    // Number of bits to shift left.
    int ss = k * 32;

    BigInteger result = vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1)
        .shiftLeft(ss).add(v0);

    if (a.signum != b.signum) {
      return result.negate();
    } else {
      return result;
    }
  }


  /**
   * Returns a slice of a BigInteger for use in Toom-Cook multiplication.
   *
   * @param lowerSize The size of the lower-order bit slices.
   * @param upperSize The size of the higher-order bit slices.
   * @param slice The index of which slice is requested, which must be a number from 0 to size-1.
   * Slice 0 is the highest-order bits, and slice size-1 are the lowest-order bits. Slice 0 may be
   * of different size than the other slices.
   * @param fullsize The size of the larger integer array, used to align slices to the appropriate
   * position when multiplying different-sized numbers.
   */
  private BigInteger getToomSlice(int lowerSize, int upperSize, int slice,
      int fullsize) {
    int start, end, sliceSize, len, offset;

    len = mag.length;
    offset = fullsize - len;

    if (slice == 0) {
      start = 0 - offset;
      end = upperSize - 1 - offset;
    } else {
      start = upperSize + (slice - 1) * lowerSize - offset;
      end = start + lowerSize - 1;
    }

    if (start < 0) {
      start = 0;
    }
    if (end < 0) {
      return ZERO;
    }

    sliceSize = (end - start) + 1;

    if (sliceSize <= 0) {
      return ZERO;
    }

    // While performing Toom-Cook, all slices are positive and
    // the sign is adjusted when the final number is composed.
    if (start == 0 && sliceSize >= len) {
      return this.abs();
    }

    int intSlice[] = new int[sliceSize];
    System.arraycopy(mag, start, intSlice, 0, sliceSize);

    return new BigInteger(trustedStripLeadingZeroInts(intSlice), 1);
  }

  /**
   * Does an exact division (that is, the remainder is known to be zero)
   * of the specified number by 3.  This is used in Toom-Cook
   * multiplication.  This is an efficient algorithm that runs in linear
   * time.  If the argument is not exactly divisible by 3, results are
   * undefined.  Note that this is expected to be called with positive
   * arguments only.
   */
  private BigInteger exactDivideBy3() {
    int len = mag.length;
    int[] result = new int[len];
    long x, w, q, borrow;
    borrow = 0L;
    for (int i = len - 1; i >= 0; i--) {
      x = (mag[i] & LONG_MASK);
      w = x - borrow;
      if (borrow > x) {      // Did we make the number go negative?
        borrow = 1L;
      } else {
        borrow = 0L;
      }

      // 0xAAAAAAAB is the modular inverse of 3 (mod 2^32).  Thus,
      // the effect of this is to divide by 3 (mod 2^32).
      // This is much faster than division on most architectures.
      q = (w * 0xAAAAAAABL) & LONG_MASK;
      result[i] = (int) q;

      // Now check the borrow. The second check can of course be
      // eliminated if the first fails.
      if (q >= 0x55555556L) {
        borrow++;
        if (q >= 0xAAAAAAABL) {
          borrow++;
        }
      }
    }
    result = trustedStripLeadingZeroInts(result);
    return new BigInteger(result, signum);
  }

  /**
   * Returns a new BigInteger representing n lower ints of the number.
   * This is used by Karatsuba multiplication and Karatsuba squaring.
   */
  private BigInteger getLower(int n) {
    int len = mag.length;

    if (len <= n) {
      return abs();
    }

    int lowerInts[] = new int[n];
    System.arraycopy(mag, len - n, lowerInts, 0, n);

    return new BigInteger(trustedStripLeadingZeroInts(lowerInts), 1);
  }

  /**
   * Returns a new BigInteger representing mag.length-n upper
   * ints of the number.  This is used by Karatsuba multiplication and
   * Karatsuba squaring.
   */
  private BigInteger getUpper(int n) {
    int len = mag.length;

    if (len <= n) {
      return ZERO;
    }

    int upperLen = len - n;
    int upperInts[] = new int[upperLen];
    System.arraycopy(mag, 0, upperInts, 0, upperLen);

    return new BigInteger(trustedStripLeadingZeroInts(upperInts), 1);
  }

  // Squaring

  /**
   * Returns a BigInteger whose value is {@code (this<sup>2</sup>)}.
   *
   * @return {@code this<sup>2</sup>}
   */
  private BigInteger square() {
    if (signum == 0) {
      return ZERO;
    }
    int len = mag.length;

    if (len < KARATSUBA_SQUARE_THRESHOLD) {
      int[] z = squareToLen(mag, len, null);
      return new BigInteger(trustedStripLeadingZeroInts(z), 1);
    } else {
      if (len < TOOM_COOK_SQUARE_THRESHOLD) {
        return squareKaratsuba();
      } else {
        return squareToomCook3();
      }
    }
  }

  /**
   * Squares the contents of the int array x. The result is placed into the
   * int array z.  The contents of x are not changed.
   */
  private static final int[] squareToLen(int[] x, int len, int[] z) {
        /*
         * The algorithm used here is adapted from Colin Plumb's C library.
         * Technique: Consider the partial products in the multiplication
         * of "abcde" by itself:
         *
         *               a  b  c  d  e
         *            *  a  b  c  d  e
         *          ==================
         *              ae be ce de ee
         *           ad bd cd dd de
         *        ac bc cc cd ce
         *     ab bb bc bd be
         *  aa ab ac ad ae
         *
         * Note that everything above the main diagonal:
         *              ae be ce de = (abcd) * e
         *           ad bd cd       = (abc) * d
         *        ac bc             = (ab) * c
         *     ab                   = (a) * b
         *
         * is a copy of everything below the main diagonal:
         *                       de
         *                 cd ce
         *           bc bd be
         *     ab ac ad ae
         *
         * Thus, the sum is 2 * (off the diagonal) + diagonal.
         *
         * This is accumulated beginning with the diagonal (which
         * consist of the squares of the digits of the input), which is then
         * divided by two, the off-diagonal added, and multiplied by two
         * again.  The low bit is simply a copy of the low bit of the
         * input, so it doesn't need special care.
         */
    int zlen = len << 1;
    if (z == null || z.length < zlen) {
      z = new int[zlen];
    }

    // Store the squares, right shifted one bit (i.e., divided by 2)
    int lastProductLowWord = 0;
    for (int j = 0, i = 0; j < len; j++) {
      long piece = (x[j] & LONG_MASK);
      long product = piece * piece;
      z[i++] = (lastProductLowWord << 31) | (int) (product >>> 33);
      z[i++] = (int) (product >>> 1);
      lastProductLowWord = (int) product;
    }

    // Add in off-diagonal sums
    for (int i = len, offset = 1; i > 0; i--, offset += 2) {
      int t = x[i - 1];
      t = mulAdd(z, x, offset, i - 1, t);
      addOne(z, offset - 1, i, t);
    }

    // Shift back up and set low bit
    primitiveLeftShift(z, zlen, 1);
    z[zlen - 1] |= x[len - 1] & 1;

    return z;
  }

  /**
   * Squares a BigInteger using the Karatsuba squaring algorithm.  It should
   * be used when both numbers are larger than a certain threshold (found
   * experimentally).  It is a recursive divide-and-conquer algorithm that
   * has better asymptotic performance than the algorithm used in
   * squareToLen.
   */
  private BigInteger squareKaratsuba() {
    int half = (mag.length + 1) / 2;

    BigInteger xl = getLower(half);
    BigInteger xh = getUpper(half);

    BigInteger xhs = xh.square();  // xhs = xh^2
    BigInteger xls = xl.square();  // xls = xl^2

    // xh^2 << 64  +  (((xl+xh)^2 - (xh^2 + xl^2)) << 32) + xl^2
    return xhs.shiftLeft(half * 32).add(xl.add(xh).square().subtract(xhs.add(xls)))
        .shiftLeft(half * 32).add(xls);
  }

  /**
   * Squares a BigInteger using the 3-way Toom-Cook squaring algorithm.  It
   * should be used when both numbers are larger than a certain threshold
   * (found experimentally).  It is a recursive divide-and-conquer algorithm
   * that has better asymptotic performance than the algorithm used in
   * squareToLen or squareKaratsuba.
   */
  private BigInteger squareToomCook3() {
    int len = mag.length;

    // k is the size (in ints) of the lower-order slices.
    int k = (len + 2) / 3;   // Equal to ceil(largest/3)

    // r is the size (in ints) of the highest-order slice.
    int r = len - 2 * k;

    // Obtain slices of the numbers. a2 is the most significant
    // bits of the number, and a0 the least significant.
    BigInteger a0, a1, a2;
    a2 = getToomSlice(k, r, 0, len);
    a1 = getToomSlice(k, r, 1, len);
    a0 = getToomSlice(k, r, 2, len);
    BigInteger v0, v1, v2, vm1, vinf, t1, t2, tm1, da1;

    v0 = a0.square();
    da1 = a2.add(a0);
    vm1 = da1.subtract(a1).square();
    da1 = da1.add(a1);
    v1 = da1.square();
    vinf = a2.square();
    v2 = da1.add(a2).shiftLeft(1).subtract(a0).square();

    // The algorithm requires two divisions by 2 and one by 3.
    // All divisions are known to be exact, that is, they do not produce
    // remainders, and all results are positive.  The divisions by 2 are
    // implemented as right shifts which are relatively efficient, leaving
    // only a division by 3.
    // The division by 3 is done by an optimized algorithm for this case.
    t2 = v2.subtract(vm1).exactDivideBy3();
    tm1 = v1.subtract(vm1).shiftRight(1);
    t1 = v1.subtract(v0);
    t2 = t2.subtract(t1).shiftRight(1);
    t1 = t1.subtract(tm1).subtract(vinf);
    t2 = t2.subtract(vinf.shiftLeft(1));
    tm1 = tm1.subtract(t2);

    // Number of bits to shift left.
    int ss = k * 32;

    return vinf.shiftLeft(ss).add(t2).shiftLeft(ss).add(t1).shiftLeft(ss).add(tm1).shiftLeft(ss)
        .add(v0);
  }

  // Division

  /**
   * Returns a BigInteger whose value is {@code (this / val)}.
   *
   * @param val value by which this BigInteger is to be divided.
   * @return {@code this / val}
   * @throws ArithmeticException if {@code val} is zero.
   */
  public BigInteger divide(BigInteger val) {
    if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
        mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
      return divideKnuth(val);
    } else {
      return divideBurnikelZiegler(val);
    }
  }

  /**
   * Returns a BigInteger whose value is {@code (this / val)} using an O(n^2) algorithm from Knuth.
   *
   * @param val value by which this BigInteger is to be divided.
   * @return {@code this / val}
   * @throws ArithmeticException if {@code val} is zero.
   * @see MutableBigInteger#divideKnuth(MutableBigInteger, MutableBigInteger, boolean)
   */
  private BigInteger divideKnuth(BigInteger val) {
    MutableBigInteger q = new MutableBigInteger(),
        a = new MutableBigInteger(this.mag),
        b = new MutableBigInteger(val.mag);

    a.divideKnuth(b, q, false);
    return q.toBigInteger(this.signum * val.signum);
  }

  /**
   * Returns an array of two BigIntegers containing {@code (this / val)}
   * followed by {@code (this % val)}.
   *
   * @param val value by which this BigInteger is to be divided, and the remainder computed.
   * @return an array of two BigIntegers: the quotient {@code (this / val)} is the initial element,
   * and the remainder {@code (this % val)} is the final element.
   * @throws ArithmeticException if {@code val} is zero.
   */
  public BigInteger[] divideAndRemainder(BigInteger val) {
    if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
        mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
      return divideAndRemainderKnuth(val);
    } else {
      return divideAndRemainderBurnikelZiegler(val);
    }
  }

  /**
   * Long division
   */
  private BigInteger[] divideAndRemainderKnuth(BigInteger val) {
    BigInteger[] result = new BigInteger[2];
    MutableBigInteger q = new MutableBigInteger(),
        a = new MutableBigInteger(this.mag),
        b = new MutableBigInteger(val.mag);
    MutableBigInteger r = a.divideKnuth(b, q);
    result[0] = q.toBigInteger(this.signum == val.signum ? 1 : -1);
    result[1] = r.toBigInteger(this.signum);
    return result;
  }

  /**
   * Returns a BigInteger whose value is {@code (this % val)}.
   *
   * @param val value by which this BigInteger is to be divided, and the remainder computed.
   * @return {@code this % val}
   * @throws ArithmeticException if {@code val} is zero.
   */
  public BigInteger remainder(BigInteger val) {
    if (val.mag.length < BURNIKEL_ZIEGLER_THRESHOLD ||
        mag.length - val.mag.length < BURNIKEL_ZIEGLER_OFFSET) {
      return remainderKnuth(val);
    } else {
      return remainderBurnikelZiegler(val);
    }
  }

  /**
   * Long division
   */
  private BigInteger remainderKnuth(BigInteger val) {
    MutableBigInteger q = new MutableBigInteger(),
        a = new MutableBigInteger(this.mag),
        b = new MutableBigInteger(val.mag);

    return a.divideKnuth(b, q).toBigInteger(this.signum);
  }

  /**
   * Calculates {@code this / val} using the Burnikel-Ziegler algorithm.
   *
   * @param val the divisor
   * @return {@code this / val}
   */
  private BigInteger divideBurnikelZiegler(BigInteger val) {
    return divideAndRemainderBurnikelZiegler(val)[0];
  }

  /**
   * Calculates {@code this % val} using the Burnikel-Ziegler algorithm.
   *
   * @param val the divisor
   * @return {@code this % val}
   */
  private BigInteger remainderBurnikelZiegler(BigInteger val) {
    return divideAndRemainderBurnikelZiegler(val)[1];
  }

  /**
   * Computes {@code this / val} and {@code this % val} using the
   * Burnikel-Ziegler algorithm.
   *
   * @param val the divisor
   * @return an array containing the quotient and remainder
   */
  private BigInteger[] divideAndRemainderBurnikelZiegler(BigInteger val) {
    MutableBigInteger q = new MutableBigInteger();
    MutableBigInteger r = new MutableBigInteger(this)
        .divideAndRemainderBurnikelZiegler(new MutableBigInteger(val), q);
    BigInteger qBigInt = q.isZero() ? ZERO : q.toBigInteger(signum * val.signum);
    BigInteger rBigInt = r.isZero() ? ZERO : r.toBigInteger(signum);
    return new BigInteger[]{qBigInt, rBigInt};
  }

  /**
   * Returns a BigInteger whose value is <tt>(this<sup>exponent</sup>)</tt>.
   * Note that {@code exponent} is an integer rather than a BigInteger.
   *
   * @param exponent exponent to which this BigInteger is to be raised.
   * @return <tt>this<sup>exponent</sup></tt>
   * @throws ArithmeticException {@code exponent} is negative.  (This would cause the operation to
   * yield a non-integer value.)
   */
  public BigInteger pow(int exponent) {
    if (exponent < 0) {
      throw new ArithmeticException("Negative exponent");
    }
    if (signum == 0) {
      return (exponent == 0 ? ONE : this);
    }

    BigInteger partToSquare = this.abs();

    // Factor out powers of two from the base, as the exponentiation of
    // these can be done by left shifts only.
    // The remaining part can then be exponentiated faster.  The
    // powers of two will be multiplied back at the end.
    int powersOfTwo = partToSquare.getLowestSetBit();
    long bitsToShift = (long) powersOfTwo * exponent;
    if (bitsToShift > Integer.MAX_VALUE) {
      reportOverflow();
    }

    int remainingBits;

    // Factor the powers of two out quickly by shifting right, if needed.
    if (powersOfTwo > 0) {
      partToSquare = partToSquare.shiftRight(powersOfTwo);
      remainingBits = partToSquare.bitLength();
      if (remainingBits == 1) {  // Nothing left but +/- 1?
        if (signum < 0 && (exponent & 1) == 1) {
          return NEGATIVE_ONE.shiftLeft(powersOfTwo * exponent);
        } else {
          return ONE.shiftLeft(powersOfTwo * exponent);
        }
      }
    } else {
      remainingBits = partToSquare.bitLength();
      if (remainingBits == 1) { // Nothing left but +/- 1?
        if (signum < 0 && (exponent & 1) == 1) {
          return NEGATIVE_ONE;
        } else {
          return ONE;
        }
      }
    }

    // This is a quick way to approximate the size of the result,
    // similar to doing log2[n] * exponent.  This will give an upper bound
    // of how big the result can be, and which algorithm to use.
    long scaleFactor = (long) remainingBits * exponent;

    // Use slightly different algorithms for small and large operands.
    // See if the result will safely fit into a long. (Largest 2^63-1)
    if (partToSquare.mag.length == 1 && scaleFactor <= 62) {
      // Small number algorithm.  Everything fits into a long.
      int newSign = (signum < 0 && (exponent & 1) == 1 ? -1 : 1);
      long result = 1;
      long baseToPow2 = partToSquare.mag[0] & LONG_MASK;

      int workingExponent = exponent;

      // Perform exponentiation using repeated squaring trick
      while (workingExponent != 0) {
        if ((workingExponent & 1) == 1) {
          result = result * baseToPow2;
        }

        if ((workingExponent >>>= 1) != 0) {
          baseToPow2 = baseToPow2 * baseToPow2;
        }
      }

      // Multiply back the powers of two (quickly, by shifting left)
      if (powersOfTwo > 0) {
        if (bitsToShift + scaleFactor <= 62) { // Fits in long?
          return valueOf((result << bitsToShift) * newSign);
        } else {
          return valueOf(result * newSign).shiftLeft((int) bitsToShift);
        }
      } else {
        return valueOf(result * newSign);
      }
    } else {
      // Large number algorithm.  This is basically identical to
      // the algorithm above, but calls multiply() and square()
      // which may use more efficient algorithms for large numbers.
      BigInteger answer = ONE;

      int workingExponent = exponent;
      // Perform exponentiation using repeated squaring trick
      while (workingExponent != 0) {
        if ((workingExponent & 1) == 1) {
          answer = answer.multiply(partToSquare);
        }

        if ((workingExponent >>>= 1) != 0) {
          partToSquare = partToSquare.square();
        }
      }
      // Multiply back the (exponentiated) powers of two (quickly,
      // by shifting left)
      if (powersOfTwo > 0) {
        answer = answer.shiftLeft(powersOfTwo * exponent);
      }

      if (signum < 0 && (exponent & 1) == 1) {
        return answer.negate();
      } else {
        return answer;
      }
    }
  }

  /**
   * Returns a BigInteger whose value is the greatest common divisor of
   * {@code abs(this)} and {@code abs(val)}.  Returns 0 if
   * {@code this == 0 && val == 0}.
   *
   * @param val value with which the GCD is to be computed.
   * @return {@code GCD(abs(this), abs(val))}
   */
  public BigInteger gcd(BigInteger val) {
    if (val.signum == 0) {
      return this.abs();
    } else if (this.signum == 0) {
      return val.abs();
    }

    MutableBigInteger a = new MutableBigInteger(this);
    MutableBigInteger b = new MutableBigInteger(val);

    MutableBigInteger result = a.hybridGCD(b);

    return result.toBigInteger(1);
  }

  /**
   * Package private method to return bit length for an integer.
   */
  static int bitLengthForInt(int n) {
    return 32 - Integer.numberOfLeadingZeros(n);
  }

  /**
   * Left shift int array a up to len by n bits. Returns the array that
   * results from the shift since space may have to be reallocated.
   */
  private static int[] leftShift(int[] a, int len, int n) {
    int nInts = n >>> 5;
    int nBits = n & 0x1F;
    int bitsInHighWord = bitLengthForInt(a[0]);

    // If shift can be done without recopy, do so
    if (n <= (32 - bitsInHighWord)) {
      primitiveLeftShift(a, len, nBits);
      return a;
    } else { // Array must be resized
      if (nBits <= (32 - bitsInHighWord)) {
        int result[] = new int[nInts + len];
        System.arraycopy(a, 0, result, 0, len);
        primitiveLeftShift(result, result.length, nBits);
        return result;
      } else {
        int result[] = new int[nInts + len + 1];
        System.arraycopy(a, 0, result, 0, len);
        primitiveRightShift(result, result.length, 32 - nBits);
        return result;
      }
    }
  }

  // shifts a up to len right n bits assumes no leading zeros, 0<n<32
  static void primitiveRightShift(int[] a, int len, int n) {
    int n2 = 32 - n;
    for (int i = len - 1, c = a[i]; i > 0; i--) {
      int b = c;
      c = a[i - 1];
      a[i] = (c << n2) | (b >>> n);
    }
    a[0] >>>= n;
  }

  // shifts a up to len left n bits assumes no leading zeros, 0<=n<32
  static void primitiveLeftShift(int[] a, int len, int n) {
    if (len == 0 || n == 0) {
      return;
    }

    int n2 = 32 - n;
    for (int i = 0, c = a[i], m = i + len - 1; i < m; i++) {
      int b = c;
      c = a[i + 1];
      a[i] = (b << n) | (c >>> n2);
    }
    a[len - 1] <<= n;
  }

  /**
   * Calculate bitlength of contents of the first len elements an int array,
   * assuming there are no leading zero ints.
   */
  private static int bitLength(int[] val, int len) {
    if (len == 0) {
      return 0;
    }
    return ((len - 1) << 5) + bitLengthForInt(val[0]);
  }

  /**
   * Returns a BigInteger whose value is the absolute value of this
   * BigInteger.
   *
   * @return {@code abs(this)}
   */
  public BigInteger abs() {
    return (signum >= 0 ? this : this.negate());
  }

  /**
   * Returns a BigInteger whose value is {@code (-this)}.
   *
   * @return {@code -this}
   */
  public BigInteger negate() {
    return new BigInteger(this.mag, -this.signum);
  }

  /**
   * Returns the signum function of this BigInteger.
   *
   * @return -1, 0 or 1 as the value of this BigInteger is negative, zero or positive.
   */
  public int signum() {
    return this.signum;
  }

  // Modular Arithmetic Operations

  /**
   * Returns a BigInteger whose value is {@code (this mod m}).  This method
   * differs from {@code remainder} in that it always returns a
   * <i>non-negative</i> BigInteger.
   *
   * @param m the modulus.
   * @return {@code this mod m}
   * @throws ArithmeticException {@code m} &le; 0
   * @see #remainder
   */
  public BigInteger mod(BigInteger m) {
    if (m.signum <= 0) {
      throw new ArithmeticException("BigInteger: modulus not positive");
    }

    BigInteger result = this.remainder(m);
    return (result.signum >= 0 ? result : result.add(m));
  }

  /**
   * Returns a BigInteger whose value is
   * <tt>(this<sup>exponent</sup> mod m)</tt>.  (Unlike {@code pow}, this
   * method permits negative exponents.)
   *
   * @param exponent the exponent.
   * @param m the modulus.
   * @return <tt>this<sup>exponent</sup> mod m</tt>
   * @throws ArithmeticException {@code m} &le; 0 or the exponent is negative and this BigInteger is
   * not <i>relatively prime</i> to {@code m}.
   * @see #modInverse
   */
  public BigInteger modPow(BigInteger exponent, BigInteger m) {
    if (m.signum <= 0) {
      throw new ArithmeticException("BigInteger: modulus not positive");
    }

    // Trivial cases
    if (exponent.signum == 0) {
      return (m.equals(ONE) ? ZERO : ONE);
    }

    if (this.equals(ONE)) {
      return (m.equals(ONE) ? ZERO : ONE);
    }

    if (this.equals(ZERO) && exponent.signum >= 0) {
      return ZERO;
    }

    if (this.equals(negConst[1]) && (!exponent.testBit(0))) {
      return (m.equals(ONE) ? ZERO : ONE);
    }

    boolean invertResult;
    if ((invertResult = (exponent.signum < 0))) {
      exponent = exponent.negate();
    }

    BigInteger base = (this.signum < 0 || this.compareTo(m) >= 0
        ? this.mod(m) : this);
    BigInteger result;
    if (m.testBit(0)) { // odd modulus
      result = base.oddModPow(exponent, m);
    } else {
            /*
             * Even modulus.  Tear it into an "odd part" (m1) and power of two
             * (m2), exponentiate mod m1, manually exponentiate mod m2, and
             * use Chinese Remainder Theorem to combine results.
             */

      // Tear m apart into odd part (m1) and power of 2 (m2)
      int p = m.getLowestSetBit();   // Max pow of 2 that divides m

      BigInteger m1 = m.shiftRight(p);  // m/2**p
      BigInteger m2 = ONE.shiftLeft(p); // 2**p

      // Calculate new base from m1
      BigInteger base2 = (this.signum < 0 || this.compareTo(m1) >= 0
          ? this.mod(m1) : this);

      // Caculate (base ** exponent) mod m1.
      BigInteger a1 = (m1.equals(ONE) ? ZERO :
          base2.oddModPow(exponent, m1));

      // Calculate (this ** exponent) mod m2
      BigInteger a2 = base.modPow2(exponent, p);

      // Combine results using Chinese Remainder Theorem
      BigInteger y1 = m2.modInverse(m1);
      BigInteger y2 = m1.modInverse(m2);

      if (m.mag.length < MAX_MAG_LENGTH / 2) {
        result = a1.multiply(m2).multiply(y1).add(a2.multiply(m1).multiply(y2)).mod(m);
      } else {
        MutableBigInteger t1 = new MutableBigInteger();
        new MutableBigInteger(a1.multiply(m2)).multiply(new MutableBigInteger(y1), t1);
        MutableBigInteger t2 = new MutableBigInteger();
        new MutableBigInteger(a2.multiply(m1)).multiply(new MutableBigInteger(y2), t2);
        t1.add(t2);
        MutableBigInteger q = new MutableBigInteger();
        result = t1.divide(new MutableBigInteger(m), q).toBigInteger();
      }
    }

    return (invertResult ? result.modInverse(m) : result);
  }

  static int[] bnExpModThreshTable = {7, 25, 81, 241, 673, 1793,
      Integer.MAX_VALUE}; // Sentinel

  /**
   * Returns a BigInteger whose value is x to the power of y mod z.
   * Assumes: z is odd && x < z.
   */
  private BigInteger oddModPow(BigInteger y, BigInteger z) {
    /*
     * The algorithm is adapted from Colin Plumb's C library.
     *
     * The window algorithm:
     * The idea is to keep a running product of b1 = n^(high-order bits of exp)
     * and then keep appending exponent bits to it.  The following patterns
     * apply to a 3-bit window (k = 3):
     * To append   0: square
     * To append   1: square, multiply by n^1
     * To append  10: square, multiply by n^1, square
     * To append  11: square, square, multiply by n^3
     * To append 100: square, multiply by n^1, square, square
     * To append 101: square, square, square, multiply by n^5
     * To append 110: square, square, multiply by n^3, square
     * To append 111: square, square, square, multiply by n^7
     *
     * Since each pattern involves only one multiply, the longer the pattern
     * the better, except that a 0 (no multiplies) can be appended directly.
     * We precompute a table of odd powers of n, up to 2^k, and can then
     * multiply k bits of exponent at a time.  Actually, assuming random
     * exponents, there is on average one zero bit between needs to
     * multiply (1/2 of the time there's none, 1/4 of the time there's 1,
     * 1/8 of the time, there's 2, 1/32 of the time, there's 3, etc.), so
     * you have to do one multiply per k+1 bits of exponent.
     *
     * The loop walks down the exponent, squaring the result buffer as
     * it goes.  There is a wbits+1 bit lookahead buffer, buf, that is
     * filled with the upcoming exponent bits.  (What is read after the
     * end of the exponent is unimportant, but it is filled with zero here.)
     * When the most-significant bit of this buffer becomes set, i.e.
     * (buf & tblmask) != 0, we have to decide what pattern to multiply
     * by, and when to do it.  We decide, remember to do it in future
     * after a suitable number of squarings have passed (e.g. a pattern
     * of "100" in the buffer requires that we multiply by n^1 immediately;
     * a pattern of "110" calls for multiplying by n^3 after one more
     * squaring), clear the buffer, and continue.
     *
     * When we start, there is one more optimization: the result buffer
     * is implcitly one, so squaring it or multiplying by it can be
     * optimized away.  Further, if we start with a pattern like "100"
     * in the lookahead window, rather than placing n into the buffer
     * and then starting to square it, we have already computed n^2
     * to compute the odd-powers table, so we can place that into
     * the buffer and save a squaring.
     *
     * This means that if you have a k-bit window, to compute n^z,
     * where z is the high k bits of the exponent, 1/2 of the time
     * it requires no squarings.  1/4 of the time, it requires 1
     * squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
     * And the remaining 1/2^(k-1) of the time, the top k bits are a
     * 1 followed by k-1 0 bits, so it again only requires k-2
     * squarings, not k-1.  The average of these is 1.  Add that
     * to the one squaring we have to do to compute the table,
     * and you'll see that a k-bit window saves k-2 squarings
     * as well as reducing the multiplies.  (It actually doesn't
     * hurt in the case k = 1, either.)
     */
    // Special case for exponent of one
    if (y.equals(ONE)) {
      return this;
    }

    // Special case for base of zero
    if (signum == 0) {
      return ZERO;
    }

    int[] base = mag.clone();
    int[] exp = y.mag;
    int[] mod = z.mag;
    int modLen = mod.length;

    // Select an appropriate window size
    int wbits = 0;
    int ebits = bitLength(exp, exp.length);
    // if exponent is 65537 (0x10001), use minimum window size
    if ((ebits != 17) || (exp[0] != 65537)) {
      while (ebits > bnExpModThreshTable[wbits]) {
        wbits++;
      }
    }

    // Calculate appropriate table size
    int tblmask = 1 << wbits;

    // Allocate table for precomputed odd powers of base in Montgomery form
    int[][] table = new int[tblmask][];
    for (int i = 0; i < tblmask; i++) {
      table[i] = new int[modLen];
    }

    // Compute the modular inverse
    int inv = -MutableBigInteger.inverseMod32(mod[modLen - 1]);

    // Convert base to Montgomery form
    int[] a = leftShift(base, base.length, modLen << 5);

    MutableBigInteger q = new MutableBigInteger(),
        a2 = new MutableBigInteger(a),
        b2 = new MutableBigInteger(mod);

    MutableBigInteger r = a2.divide(b2, q);
    table[0] = r.toIntArray();

    // Pad table[0] with leading zeros so its length is at least modLen
    if (table[0].length < modLen) {
      int offset = modLen - table[0].length;
      int[] t2 = new int[modLen];
      for (int i = 0; i < table[0].length; i++) {
        t2[i + offset] = table[0][i];
      }
      table[0] = t2;
    }

    // Set b to the square of the base
    int[] b = squareToLen(table[0], modLen, null);
    b = montReduce(b, mod, modLen, inv);

    // Set t to high half of b
    int[] t = Arrays.copyOf(b, modLen);

    // Fill in the table with odd powers of the base
    for (int i = 1; i < tblmask; i++) {
      int[] prod = multiplyToLen(t, modLen, table[i - 1], modLen, null);
      table[i] = montReduce(prod, mod, modLen, inv);
    }

    // Pre load the window that slides over the exponent
    int bitpos = 1 << ((ebits - 1) & (32 - 1));

    int buf = 0;
    int elen = exp.length;
    int eIndex = 0;
    for (int i = 0; i <= wbits; i++) {
      buf = (buf << 1) | (((exp[eIndex] & bitpos) != 0) ? 1 : 0);
      bitpos >>>= 1;
      if (bitpos == 0) {
        eIndex++;
        bitpos = 1 << (32 - 1);
        elen--;
      }
    }

    int multpos = ebits;

    // The first iteration, which is hoisted out of the main loop
    ebits--;
    boolean isone = true;

    multpos = ebits - wbits;
    while ((buf & 1) == 0) {
      buf >>>= 1;
      multpos++;
    }

    int[] mult = table[buf >>> 1];

    buf = 0;
    if (multpos == ebits) {
      isone = false;
    }

    // The main loop
    while (true) {
      ebits--;
      // Advance the window
      buf <<= 1;

      if (elen != 0) {
        buf |= ((exp[eIndex] & bitpos) != 0) ? 1 : 0;
        bitpos >>>= 1;
        if (bitpos == 0) {
          eIndex++;
          bitpos = 1 << (32 - 1);
          elen--;
        }
      }

      // Examine the window for pending multiplies
      if ((buf & tblmask) != 0) {
        multpos = ebits - wbits;
        while ((buf & 1) == 0) {
          buf >>>= 1;
          multpos++;
        }
        mult = table[buf >>> 1];
        buf = 0;
      }

      // Perform multiply
      if (ebits == multpos) {
        if (isone) {
          b = mult.clone();
          isone = false;
        } else {
          t = b;
          a = multiplyToLen(t, modLen, mult, modLen, a);
          a = montReduce(a, mod, modLen, inv);
          t = a;
          a = b;
          b = t;
        }
      }

      // Check if done
      if (ebits == 0) {
        break;
      }

      // Square the input
      if (!isone) {
        t = b;
        a = squareToLen(t, modLen, a);
        a = montReduce(a, mod, modLen, inv);
        t = a;
        a = b;
        b = t;
      }
    }

    // Convert result out of Montgomery form and return
    int[] t2 = new int[2 * modLen];
    System.arraycopy(b, 0, t2, modLen, modLen);

    b = montReduce(t2, mod, modLen, inv);

    t2 = Arrays.copyOf(b, modLen);

    return new BigInteger(1, t2);
  }

  /**
   * Montgomery reduce n, modulo mod.  This reduces modulo mod and divides
   * by 2^(32*mlen). Adapted from Colin Plumb's C library.
   */
  private static int[] montReduce(int[] n, int[] mod, int mlen, int inv) {
    int c = 0;
    int len = mlen;
    int offset = 0;

    do {
      int nEnd = n[n.length - 1 - offset];
      int carry = mulAdd(n, mod, offset, mlen, inv * nEnd);
      c += addOne(n, offset, mlen, carry);
      offset++;
    } while (--len > 0);

    while (c > 0) {
      c += subN(n, mod, mlen);
    }

    while (intArrayCmpToLen(n, mod, mlen) >= 0) {
      subN(n, mod, mlen);
    }

    return n;
  }


  /*
     * Returns -1, 0 or +1 as big-endian unsigned int array arg1 is less than,
     * equal to, or greater than arg2 up to length len.
     */
  private static int intArrayCmpToLen(int[] arg1, int[] arg2, int len) {
    for (int i = 0; i < len; i++) {
      long b1 = arg1[i] & LONG_MASK;
      long b2 = arg2[i] & LONG_MASK;
      if (b1 < b2) {
        return -1;
      }
      if (b1 > b2) {
        return 1;
      }
    }
    return 0;
  }

  /**
   * Subtracts two numbers of same length, returning borrow.
   */
  private static int subN(int[] a, int[] b, int len) {
    long sum = 0;

    while (--len >= 0) {
      sum = (a[len] & LONG_MASK) -
          (b[len] & LONG_MASK) + (sum >> 32);
      a[len] = (int) sum;
    }

    return (int) (sum >> 32);
  }

  /**
   * Multiply an array by one word k and add to result, return the carry
   */
  static int mulAdd(int[] out, int[] in, int offset, int len, int k) {
    long kLong = k & LONG_MASK;
    long carry = 0;

    offset = out.length - offset - 1;
    for (int j = len - 1; j >= 0; j--) {
      long product = (in[j] & LONG_MASK) * kLong +
          (out[offset] & LONG_MASK) + carry;
      out[offset--] = (int) product;
      carry = product >>> 32;
    }
    return (int) carry;
  }

  /**
   * Add one word to the number a mlen words into a. Return the resulting
   * carry.
   */
  static int addOne(int[] a, int offset, int mlen, int carry) {
    offset = a.length - 1 - mlen - offset;
    long t = (a[offset] & LONG_MASK) + (carry & LONG_MASK);

    a[offset] = (int) t;
    if ((t >>> 32) == 0) {
      return 0;
    }
    while (--mlen >= 0) {
      if (--offset < 0) { // Carry out of number
        return 1;
      } else {
        a[offset]++;
        if (a[offset] != 0) {
          return 0;
        }
      }
    }
    return 1;
  }

  /**
   * Returns a BigInteger whose value is (this ** exponent) mod (2**p)
   */
  private BigInteger modPow2(BigInteger exponent, int p) {
        /*
         * Perform exponentiation using repeated squaring trick, chopping off
         * high order bits as indicated by modulus.
         */
    BigInteger result = ONE;
    BigInteger baseToPow2 = this.mod2(p);
    int expOffset = 0;

    int limit = exponent.bitLength();

    if (this.testBit(0)) {
      limit = (p - 1) < limit ? (p - 1) : limit;
    }

    while (expOffset < limit) {
      if (exponent.testBit(expOffset)) {
        result = result.multiply(baseToPow2).mod2(p);
      }
      expOffset++;
      if (expOffset < limit) {
        baseToPow2 = baseToPow2.square().mod2(p);
      }
    }

    return result;
  }

  /**
   * Returns a BigInteger whose value is this mod(2**p).
   * Assumes that this {@code BigInteger >= 0} and {@code p > 0}.
   */
  private BigInteger mod2(int p) {
    if (bitLength() <= p) {
      return this;
    }

    // Copy remaining ints of mag
    int numInts = (p + 31) >>> 5;
    int[] mag = new int[numInts];
    System.arraycopy(this.mag, (this.mag.length - numInts), mag, 0, numInts);

    // Mask out any excess bits
    int excessBits = (numInts << 5) - p;
    mag[0] &= (1L << (32 - excessBits)) - 1;

    return (mag[0] == 0 ? new BigInteger(1, mag) : new BigInteger(mag, 1));
  }

  /**
   * Returns a BigInteger whose value is {@code (this}<sup>-1</sup> {@code mod m)}.
   *
   * @param m the modulus.
   * @return {@code this}<sup>-1</sup> {@code mod m}.
   * @throws ArithmeticException {@code  m} &le; 0, or this BigInteger has no multiplicative inverse
   * mod m (that is, this BigInteger is not <i>relatively prime</i> to m).
   */
  public BigInteger modInverse(BigInteger m) {
    if (m.signum != 1) {
      throw new ArithmeticException("BigInteger: modulus not positive");
    }

    if (m.equals(ONE)) {
      return ZERO;
    }

    // Calculate (this mod m)
    BigInteger modVal = this;
    if (signum < 0 || (this.compareMagnitude(m) >= 0)) {
      modVal = this.mod(m);
    }

    if (modVal.equals(ONE)) {
      return ONE;
    }

    MutableBigInteger a = new MutableBigInteger(modVal);
    MutableBigInteger b = new MutableBigInteger(m);

    MutableBigInteger result = a.mutableModInverse(b);
    return result.toBigInteger(1);
  }

  // Shift Operations

  /**
   * Returns a BigInteger whose value is {@code (this << n)}.
   * The shift distance, {@code n}, may be negative, in which case
   * this method performs a right shift.
   * (Computes <tt>floor(this * 2<sup>n</sup>)</tt>.)
   *
   * @param n shift distance, in bits.
   * @return {@code this << n}
   * @see #shiftRight
   */
  public BigInteger shiftLeft(int n) {
    if (signum == 0) {
      return ZERO;
    }
    if (n > 0) {
      return new BigInteger(shiftLeft(mag, n), signum);
    } else if (n == 0) {
      return this;
    } else {
      // Possible int overflow in (-n) is not a trouble,
      // because shiftRightImpl considers its argument unsigned
      return shiftRightImpl(-n);
    }
  }

  /**
   * Returns a magnitude array whose value is {@code (mag << n)}.
   * The shift distance, {@code n}, is considered unnsigned.
   * (Computes <tt>this * 2<sup>n</sup></tt>.)
   *
   * @param mag magnitude, the most-significant int ({@code mag[0]}) must be non-zero.
   * @param n unsigned shift distance, in bits.
   * @return {@code mag << n}
   */
  private static int[] shiftLeft(int[] mag, int n) {
    int nInts = n >>> 5;
    int nBits = n & 0x1f;
    int magLen = mag.length;
    int newMag[] = null;

    if (nBits == 0) {
      newMag = new int[magLen + nInts];
      System.arraycopy(mag, 0, newMag, 0, magLen);
    } else {
      int i = 0;
      int nBits2 = 32 - nBits;
      int highBits = mag[0] >>> nBits2;
      if (highBits != 0) {
        newMag = new int[magLen + nInts + 1];
        newMag[i++] = highBits;
      } else {
        newMag = new int[magLen + nInts];
      }
      int j = 0;
      while (j < magLen - 1) {
        newMag[i++] = mag[j++] << nBits | mag[j] >>> nBits2;
      }
      newMag[i] = mag[j] << nBits;
    }
    return newMag;
  }

  /**
   * Returns a BigInteger whose value is {@code (this >> n)}.  Sign
   * extension is performed.  The shift distance, {@code n}, may be
   * negative, in which case this method performs a left shift.
   * (Computes <tt>floor(this / 2<sup>n</sup>)</tt>.)
   *
   * @param n shift distance, in bits.
   * @return {@code this >> n}
   * @see #shiftLeft
   */
  public BigInteger shiftRight(int n) {
    if (signum == 0) {
      return ZERO;
    }
    if (n > 0) {
      return shiftRightImpl(n);
    } else if (n == 0) {
      return this;
    } else {
      // Possible int overflow in {@code -n} is not a trouble,
      // because shiftLeft considers its argument unsigned
      return new BigInteger(shiftLeft(mag, -n), signum);
    }
  }

  /**
   * Returns a BigInteger whose value is {@code (this >> n)}. The shift
   * distance, {@code n}, is considered unsigned.
   * (Computes <tt>floor(this * 2<sup>-n</sup>)</tt>.)
   *
   * @param n unsigned shift distance, in bits.
   * @return {@code this >> n}
   */
  private BigInteger shiftRightImpl(int n) {
    int nInts = n >>> 5;
    int nBits = n & 0x1f;
    int magLen = mag.length;
    int newMag[] = null;

    // Special case: entire contents shifted off the end
    if (nInts >= magLen) {
      return (signum >= 0 ? ZERO : negConst[1]);
    }

    if (nBits == 0) {
      int newMagLen = magLen - nInts;
      newMag = Arrays.copyOf(mag, newMagLen);
    } else {
      int i = 0;
      int highBits = mag[0] >>> nBits;
      if (highBits != 0) {
        newMag = new int[magLen - nInts];
        newMag[i++] = highBits;
      } else {
        newMag = new int[magLen - nInts - 1];
      }

      int nBits2 = 32 - nBits;
      int j = 0;
      while (j < magLen - nInts - 1) {
        newMag[i++] = (mag[j++] << nBits2) | (mag[j] >>> nBits);
      }
    }

    if (signum < 0) {
      // Find out whether any one-bits were shifted off the end.
      boolean onesLost = false;
      for (int i = magLen - 1, j = magLen - nInts; i >= j && !onesLost; i--) {
        onesLost = (mag[i] != 0);
      }
      if (!onesLost && nBits != 0) {
        onesLost = (mag[magLen - nInts - 1] << (32 - nBits) != 0);
      }

      if (onesLost) {
        newMag = javaIncrement(newMag);
      }
    }

    return new BigInteger(newMag, signum);
  }

  int[] javaIncrement(int[] val) {
    int lastSum = 0;
    for (int i = val.length - 1; i >= 0 && lastSum == 0; i--) {
      lastSum = (val[i] += 1);
    }
    if (lastSum == 0) {
      val = new int[val.length + 1];
      val[0] = 1;
    }
    return val;
  }

  // Bitwise Operations

  /**
   * Returns a BigInteger whose value is {@code (this & val)}.  (This
   * method returns a negative BigInteger if and only if this and val are
   * both negative.)
   *
   * @param val value to be AND'ed with this BigInteger.
   * @return {@code this & val}
   */
  public BigInteger and(BigInteger val) {
    int[] result = new int[Math.max(intLength(), val.intLength())];
    for (int i = 0; i < result.length; i++) {
      result[i] = (getInt(result.length - i - 1)
          & val.getInt(result.length - i - 1));
    }

    return valueOf(result);
  }

  /**
   * Returns a BigInteger whose value is {@code (this | val)}.  (This method
   * returns a negative BigInteger if and only if either this or val is
   * negative.)
   *
   * @param val value to be OR'ed with this BigInteger.
   * @return {@code this | val}
   */
  public BigInteger or(BigInteger val) {
    int[] result = new int[Math.max(intLength(), val.intLength())];
    for (int i = 0; i < result.length; i++) {
      result[i] = (getInt(result.length - i - 1)
          | val.getInt(result.length - i - 1));
    }

    return valueOf(result);
  }

  /**
   * Returns a BigInteger whose value is {@code (this ^ val)}.  (This method
   * returns a negative BigInteger if and only if exactly one of this and
   * val are negative.)
   *
   * @param val value to be XOR'ed with this BigInteger.
   * @return {@code this ^ val}
   */
  public BigInteger xor(BigInteger val) {
    int[] result = new int[Math.max(intLength(), val.intLength())];
    for (int i = 0; i < result.length; i++) {
      result[i] = (getInt(result.length - i - 1)
          ^ val.getInt(result.length - i - 1));
    }

    return valueOf(result);
  }

  /**
   * Returns a BigInteger whose value is {@code (~this)}.  (This method
   * returns a negative value if and only if this BigInteger is
   * non-negative.)
   *
   * @return {@code ~this}
   */
  public BigInteger not() {
    int[] result = new int[intLength()];
    for (int i = 0; i < result.length; i++) {
      result[i] = ~getInt(result.length - i - 1);
    }

    return valueOf(result);
  }

  /**
   * Returns a BigInteger whose value is {@code (this & ~val)}.  This
   * method, which is equivalent to {@code and(val.not())}, is provided as
   * a convenience for masking operations.  (This method returns a negative
   * BigInteger if and only if {@code this} is negative and {@code val} is
   * positive.)
   *
   * @param val value to be complemented and AND'ed with this BigInteger.
   * @return {@code this & ~val}
   */
  public BigInteger andNot(BigInteger val) {
    int[] result = new int[Math.max(intLength(), val.intLength())];
    for (int i = 0; i < result.length; i++) {
      result[i] = (getInt(result.length - i - 1)
          & ~val.getInt(result.length - i - 1));
    }

    return valueOf(result);
  }

  // Single Bit Operations

  /**
   * Returns {@code true} if and only if the designated bit is set.
   * (Computes {@code ((this & (1<<n)) != 0)}.)
   *
   * @param n index of bit to test.
   * @return {@code true} if and only if the designated bit is set.
   * @throws ArithmeticException {@code n} is negative.
   */
  public boolean testBit(int n) {
    if (n < 0) {
      throw new ArithmeticException("Negative bit address");
    }

    return (getInt(n >>> 5) & (1 << (n & 31))) != 0;
  }

  /**
   * Returns a BigInteger whose value is equivalent to this BigInteger
   * with the designated bit set.  (Computes {@code (this | (1<<n))}.)
   *
   * @param n index of bit to set.
   * @return {@code this | (1<<n)}
   * @throws ArithmeticException {@code n} is negative.
   */
  public BigInteger setBit(int n) {
    if (n < 0) {
      throw new ArithmeticException("Negative bit address");
    }

    int intNum = n >>> 5;
    int[] result = new int[Math.max(intLength(), intNum + 2)];

    for (int i = 0; i < result.length; i++) {
      result[result.length - i - 1] = getInt(i);
    }

    result[result.length - intNum - 1] |= (1 << (n & 31));

    return valueOf(result);
  }

  /**
   * Returns a BigInteger whose value is equivalent to this BigInteger
   * with the designated bit cleared.
   * (Computes {@code (this & ~(1<<n))}.)
   *
   * @param n index of bit to clear.
   * @return {@code this & ~(1<<n)}
   * @throws ArithmeticException {@code n} is negative.
   */
  public BigInteger clearBit(int n) {
    if (n < 0) {
      throw new ArithmeticException("Negative bit address");
    }

    int intNum = n >>> 5;
    int[] result = new int[Math.max(intLength(), ((n + 1) >>> 5) + 1)];

    for (int i = 0; i < result.length; i++) {
      result[result.length - i - 1] = getInt(i);
    }

    result[result.length - intNum - 1] &= ~(1 << (n & 31));

    return valueOf(result);
  }

  /**
   * Returns a BigInteger whose value is equivalent to this BigInteger
   * with the designated bit flipped.
   * (Computes {@code (this ^ (1<<n))}.)
   *
   * @param n index of bit to flip.
   * @return {@code this ^ (1<<n)}
   * @throws ArithmeticException {@code n} is negative.
   */
  public BigInteger flipBit(int n) {
    if (n < 0) {
      throw new ArithmeticException("Negative bit address");
    }

    int intNum = n >>> 5;
    int[] result = new int[Math.max(intLength(), intNum + 2)];

    for (int i = 0; i < result.length; i++) {
      result[result.length - i - 1] = getInt(i);
    }

    result[result.length - intNum - 1] ^= (1 << (n & 31));

    return valueOf(result);
  }

  /**
   * Returns the index of the rightmost (lowest-order) one bit in this
   * BigInteger (the number of zero bits to the right of the rightmost
   * one bit).  Returns -1 if this BigInteger contains no one bits.
   * (Computes {@code (this == 0? -1 : log2(this & -this))}.)
   *
   * @return index of the rightmost one bit in this BigInteger.
   */
  public int getLowestSetBit() {
    @SuppressWarnings("deprecation") int lsb = lowestSetBit - 2;
    if (lsb == -2) {  // lowestSetBit not initialized yet
      lsb = 0;
      if (signum == 0) {
        lsb -= 1;
      } else {
        // Search for lowest order nonzero int
        int i, b;
        for (i = 0; (b = getInt(i)) == 0; i++) {
          ;
        }
        lsb += (i << 5) + Integer.numberOfTrailingZeros(b);
      }
      lowestSetBit = lsb + 2;
    }
    return lsb;
  }

  // Miscellaneous Bit Operations

  /**
   * Returns the number of bits in the minimal two's-complement
   * representation of this BigInteger, <i>excluding</i> a sign bit.
   * For positive BigIntegers, this is equivalent to the number of bits in
   * the ordinary binary representation.  (Computes
   * {@code (ceil(log2(this < 0 ? -this : this+1)))}.)
   *
   * @return number of bits in the minimal two's-complement representation of this BigInteger,
   * <i>excluding</i> a sign bit.
   */
  public int bitLength() {
    @SuppressWarnings("deprecation") int n = bitLength - 1;
    if (n == -1) { // bitLength not initialized yet
      int[] m = mag;
      int len = m.length;
      if (len == 0) {
        n = 0; // offset by one to initialize
      } else {
        // Calculate the bit length of the magnitude
        int magBitLength = ((len - 1) << 5) + bitLengthForInt(mag[0]);
        if (signum < 0) {
          // Check if magnitude is a power of two
          boolean pow2 = (Integer.bitCount(mag[0]) == 1);
          for (int i = 1; i < len && pow2; i++) {
            pow2 = (mag[i] == 0);
          }

          n = (pow2 ? magBitLength - 1 : magBitLength);
        } else {
          n = magBitLength;
        }
      }
      bitLength = n + 1;
    }
    return n;
  }

  /**
   * Returns the number of bits in the two's complement representation
   * of this BigInteger that differ from its sign bit.  This method is
   * useful when implementing bit-vector style sets atop BigIntegers.
   *
   * @return number of bits in the two's complement representation of this BigInteger that differ
   * from its sign bit.
   */
  public int bitCount() {
    @SuppressWarnings("deprecation") int bc = bitCount - 1;
    if (bc == -1) {  // bitCount not initialized yet
      bc = 0;      // offset by one to initialize
      // Count the bits in the magnitude
      for (int i = 0; i < mag.length; i++) {
        bc += Integer.bitCount(mag[i]);
      }
      if (signum < 0) {
        // Count the trailing zeros in the magnitude
        int magTrailingZeroCount = 0, j;
        for (j = mag.length - 1; mag[j] == 0; j--) {
          magTrailingZeroCount += 32;
        }
        magTrailingZeroCount += Integer.numberOfTrailingZeros(mag[j]);
        bc += magTrailingZeroCount - 1;
      }
      bitCount = bc + 1;
    }
    return bc;
  }

  // Primality Testing

  /**
   * Returns {@code true} if this BigInteger is probably prime,
   * {@code false} if it's definitely composite.  If
   * {@code certainty} is &le; 0, {@code true} is
   * returned.
   *
   * @param certainty a measure of the uncertainty that the caller is willing to tolerate: if the
   * call returns {@code true} the probability that this BigInteger is prime exceeds (1 -
   * 1/2<sup>{@code certainty}</sup>).  The execution time of this method is proportional to the
   * value of this parameter.
   * @return {@code true} if this BigInteger is probably prime, {@code false} if it's definitely
   * composite.
   */
  public boolean isProbablePrime(int certainty) {
    if (certainty <= 0) {
      return true;
    }
    BigInteger w = this.abs();
    if (w.equals(TWO)) {
      return true;
    }
    if (!w.testBit(0) || w.equals(ONE)) {
      return false;
    }

    return w.primeToCertainty(certainty, null);
  }

  // Comparison Operations

  /**
   * Compares this BigInteger with the specified BigInteger.  This
   * method is provided in preference to individual methods for each
   * of the six boolean comparison operators ({@literal <}, ==,
   * {@literal >}, {@literal >=}, !=, {@literal <=}).  The suggested
   * idiom for performing these comparisons is: {@code
   * (x.compareTo(y)} &lt;<i>op</i>&gt; {@code 0)}, where
   * &lt;<i>op</i>&gt; is one of the six comparison operators.
   *
   * @param val BigInteger to which this BigInteger is to be compared.
   * @return -1, 0 or 1 as this BigInteger is numerically less than, equal to, or greater than
   * {@code val}.
   */
  public int compareTo(BigInteger val) {
    if (signum == val.signum) {
      switch (signum) {
        case 1:
          return compareMagnitude(val);
        case -1:
          return val.compareMagnitude(this);
        default:
          return 0;
      }
    }
    return signum > val.signum ? 1 : -1;
  }

  /**
   * Compares the magnitude array of this BigInteger with the specified
   * BigInteger's. This is the version of compareTo ignoring sign.
   *
   * @param val BigInteger whose magnitude array to be compared.
   * @return -1, 0 or 1 as this magnitude array is less than, equal to or greater than the magnitude
   * aray for the specified BigInteger's.
   */
  final int compareMagnitude(BigInteger val) {
    int[] m1 = mag;
    int len1 = m1.length;
    int[] m2 = val.mag;
    int len2 = m2.length;
    if (len1 < len2) {
      return -1;
    }
    if (len1 > len2) {
      return 1;
    }
    for (int i = 0; i < len1; i++) {
      int a = m1[i];
      int b = m2[i];
      if (a != b) {
        return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
      }
    }
    return 0;
  }

  /**
   * Version of compareMagnitude that compares magnitude with long value.
   * val can't be Long.MIN_VALUE.
   */
  final int compareMagnitude(long val) {
    assert val != Long.MIN_VALUE;
    int[] m1 = mag;
    int len = m1.length;
    if (len > 2) {
      return 1;
    }
    if (val < 0) {
      val = -val;
    }
    int highWord = (int) (val >>> 32);
    if (highWord == 0) {
      if (len < 1) {
        return -1;
      }
      if (len > 1) {
        return 1;
      }
      int a = m1[0];
      int b = (int) val;
      if (a != b) {
        return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
      }
      return 0;
    } else {
      if (len < 2) {
        return -1;
      }
      int a = m1[0];
      int b = highWord;
      if (a != b) {
        return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
      }
      a = m1[1];
      b = (int) val;
      if (a != b) {
        return ((a & LONG_MASK) < (b & LONG_MASK)) ? -1 : 1;
      }
      return 0;
    }
  }

  /**
   * Compares this BigInteger with the specified Object for equality.
   *
   * @param x Object to which this BigInteger is to be compared.
   * @return {@code true} if and only if the specified Object is a BigInteger whose value is
   * numerically equal to this BigInteger.
   */
  public boolean equals(Object x) {
    // This test is just an optimization, which may or may not help
    if (x == this) {
      return true;
    }

    if (!(x instanceof BigInteger)) {
      return false;
    }

    BigInteger xInt = (BigInteger) x;
    if (xInt.signum != signum) {
      return false;
    }

    int[] m = mag;
    int len = m.length;
    int[] xm = xInt.mag;
    if (len != xm.length) {
      return false;
    }

    for (int i = 0; i < len; i++) {
      if (xm[i] != m[i]) {
        return false;
      }
    }

    return true;
  }

  /**
   * Returns the minimum of this BigInteger and {@code val}.
   *
   * @param val value with which the minimum is to be computed.
   * @return the BigInteger whose value is the lesser of this BigInteger and {@code val}.  If they
   * are equal, either may be returned.
   */
  public BigInteger min(BigInteger val) {
    return (compareTo(val) < 0 ? this : val);
  }

  /**
   * Returns the maximum of this BigInteger and {@code val}.
   *
   * @param val value with which the maximum is to be computed.
   * @return the BigInteger whose value is the greater of this and {@code val}.  If they are equal,
   * either may be returned.
   */
  public BigInteger max(BigInteger val) {
    return (compareTo(val) > 0 ? this : val);
  }

  // Hash Function

  /**
   * Returns the hash code for this BigInteger.
   *
   * @return hash code for this BigInteger.
   */
  public int hashCode() {
    int hashCode = 0;

    for (int i = 0; i < mag.length; i++) {
      hashCode = (int) (31 * hashCode + (mag[i] & LONG_MASK));
    }

    return hashCode * signum;
  }

  /**
   * Returns the String representation of this BigInteger in the
   * given radix.  If the radix is outside the range from {@link
   * Character#MIN_RADIX} to {@link Character#MAX_RADIX} inclusive,
   * it will default to 10 (as is the case for
   * {@code Integer.toString}).  The digit-to-character mapping
   * provided by {@code Character.forDigit} is used, and a minus
   * sign is prepended if appropriate.  (This representation is
   * compatible with the {@link #BigInteger(String, int) (String,
   * int)} constructor.)
   *
   * @param radix radix of the String representation.
   * @return String representation of this BigInteger in the given radix.
   * @see Integer#toString
   * @see Character#forDigit
   * @see #BigInteger(java.lang.String, int)
   */
  public String toString(int radix) {
    if (signum == 0) {
      return "0";
    }
    if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX) {
      radix = 10;
    }

    // If it's small enough, use smallToString.
    if (mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
      return smallToString(radix);
    }

    // Otherwise use recursive toString, which requires positive arguments.
    // The results will be concatenated into this StringBuilder
    StringBuilder sb = new StringBuilder();
    if (signum < 0) {
      toString(this.negate(), sb, radix, 0);
      sb.insert(0, '-');
    } else {
      toString(this, sb, radix, 0);
    }

    return sb.toString();
  }

  /**
   * This method is used to perform toString when arguments are small.
   */
  private String smallToString(int radix) {
    if (signum == 0) {
      return "0";
    }

    // Compute upper bound on number of digit groups and allocate space
    int maxNumDigitGroups = (4 * mag.length + 6) / 7;
    String digitGroup[] = new String[maxNumDigitGroups];

    // Translate number to string, a digit group at a time
    BigInteger tmp = this.abs();
    int numGroups = 0;
    while (tmp.signum != 0) {
      BigInteger d = longRadix[radix];

      MutableBigInteger q = new MutableBigInteger(),
          a = new MutableBigInteger(tmp.mag),
          b = new MutableBigInteger(d.mag);
      MutableBigInteger r = a.divide(b, q);
      BigInteger q2 = q.toBigInteger(tmp.signum * d.signum);
      BigInteger r2 = r.toBigInteger(tmp.signum * d.signum);

      digitGroup[numGroups++] = Long.toString(r2.longValue(), radix);
      tmp = q2;
    }

    // Put sign (if any) and first digit group into result buffer
    StringBuilder buf = new StringBuilder(numGroups * digitsPerLong[radix] + 1);
    if (signum < 0) {
      buf.append('-');
    }
    buf.append(digitGroup[numGroups - 1]);

    // Append remaining digit groups padded with leading zeros
    for (int i = numGroups - 2; i >= 0; i--) {
      // Prepend (any) leading zeros for this digit group
      int numLeadingZeros = digitsPerLong[radix] - digitGroup[i].length();
      if (numLeadingZeros != 0) {
        buf.append(zeros[numLeadingZeros]);
      }
      buf.append(digitGroup[i]);
    }
    return buf.toString();
  }

  /**
   * Converts the specified BigInteger to a string and appends to
   * {@code sb}.  This implements the recursive Schoenhage algorithm
   * for base conversions.
   * <p/>
   * See Knuth, Donald,  _The Art of Computer Programming_, Vol. 2,
   * Answers to Exercises (4.4) Question 14.
   *
   * @param u The number to convert to a string.
   * @param sb The StringBuilder that will be appended to in place.
   * @param radix The base to convert to.
   * @param digits The minimum number of digits to pad to.
   */
  private static void toString(BigInteger u, StringBuilder sb, int radix,
      int digits) {
        /* If we're smaller than a certain threshold, use the smallToString
           method, padding with leading zeroes when necessary. */
    if (u.mag.length <= SCHOENHAGE_BASE_CONVERSION_THRESHOLD) {
      String s = u.smallToString(radix);

      // Pad with internal zeros if necessary.
      // Don't pad if we're at the beginning of the string.
      if ((s.length() < digits) && (sb.length() > 0)) {
        for (int i = s.length(); i < digits; i++) { // May be a faster way to
          sb.append('0');                    // do this?
        }
      }

      sb.append(s);
      return;
    }

    int b, n;
    b = u.bitLength();

    // Calculate a value for n in the equation radix^(2^n) = u
    // and subtract 1 from that value.  This is used to find the
    // cache index that contains the best value to divide u.
    n = (int) Math.round(Math.log(b * LOG_TWO / logCache[radix]) / LOG_TWO - 1.0);
    BigInteger v = getRadixConversionCache(radix, n);
    BigInteger[] results;
    results = u.divideAndRemainder(v);

    int expectedDigits = 1 << n;

    // Now recursively build the two halves of each number.
    toString(results[0], sb, radix, digits - expectedDigits);
    toString(results[1], sb, radix, expectedDigits);
  }

  /**
   * Returns the value radix^(2^exponent) from the cache.
   * If this value doesn't already exist in the cache, it is added.
   * <p/>
   * This could be changed to a more complicated caching method using
   * {@code Future}.
   */
  private static BigInteger getRadixConversionCache(int radix, int exponent) {
    BigInteger[] cacheLine = powerCache[radix]; // volatile read
    if (exponent < cacheLine.length) {
      return cacheLine[exponent];
    }

    int oldLength = cacheLine.length;
    cacheLine = Arrays.copyOf(cacheLine, exponent + 1);
    for (int i = oldLength; i <= exponent; i++) {
      cacheLine[i] = cacheLine[i - 1].pow(2);
    }

    BigInteger[][] pc = powerCache; // volatile read again
    if (exponent >= pc[radix].length) {
      pc = pc.clone();
      pc[radix] = cacheLine;
      powerCache = pc; // volatile write, publish
    }
    return cacheLine[exponent];
  }

  /* zero[i] is a string of i consecutive zeros. */
  private static String zeros[] = new String[64];

  static {
    zeros[63] =
        "000000000000000000000000000000000000000000000000000000000000000";
    for (int i = 0; i < 63; i++) {
      zeros[i] = zeros[63].substring(0, i);
    }
  }

  /**
   * Returns the decimal String representation of this BigInteger.
   * The digit-to-character mapping provided by
   * {@code Character.forDigit} is used, and a minus sign is
   * prepended if appropriate.  (This representation is compatible
   * with the {@link #BigInteger(String) (String)} constructor, and
   * allows for String concatenation with Java's + operator.)
   *
   * @return decimal String representation of this BigInteger.
   * @see Character#forDigit
   * @see #BigInteger(java.lang.String)
   */
  public String toString() {
    return toString(10);
  }

  /**
   * Returns a byte array containing the two's-complement
   * representation of this BigInteger.  The byte array will be in
   * <i>big-endian</i> byte-order: the most significant byte is in
   * the zeroth element.  The array will contain the minimum number
   * of bytes required to represent this BigInteger, including at
   * least one sign bit, which is {@code (ceil((this.bitLength() +
   * 1)/8))}.  (This representation is compatible with the
   * {@link #BigInteger(byte[]) (byte[])} constructor.)
   *
   * @return a byte array containing the two's-complement representation of this BigInteger.
   * @see #BigInteger(byte[])
   */
  public byte[] toByteArray() {
    int byteLen = bitLength() / 8 + 1;
    byte[] byteArray = new byte[byteLen];

    for (int i = byteLen - 1, bytesCopied = 4, nextInt = 0, intIndex = 0; i >= 0; i--) {
      if (bytesCopied == 4) {
        nextInt = getInt(intIndex++);
        bytesCopied = 1;
      } else {
        nextInt >>>= 8;
        bytesCopied++;
      }
      byteArray[i] = (byte) nextInt;
    }
    return byteArray;
  }

  /**
   * Converts this BigInteger to an {@code int}.  This
   * conversion is analogous to a
   * <i>narrowing primitive conversion</i> from {@code long} to
   * {@code int} as defined in section 5.1.3 of
   * <cite>The Java&trade; Language Specification</cite>:
   * if this BigInteger is too big to fit in an
   * {@code int}, only the low-order 32 bits are returned.
   * Note that this conversion can lose information about the
   * overall magnitude of the BigInteger value as well as return a
   * result with the opposite sign.
   *
   * @return this BigInteger converted to an {@code int}.
   * @see #intValueExact()
   */
  public int intValue() {
    int result = 0;
    result = getInt(0);
    return result;
  }

  /**
   * Converts this BigInteger to a {@code long}.  This
   * conversion is analogous to a
   * <i>narrowing primitive conversion</i> from {@code long} to
   * {@code int} as defined in section 5.1.3 of
   * <cite>The Java&trade; Language Specification</cite>:
   * if this BigInteger is too big to fit in a
   * {@code long}, only the low-order 64 bits are returned.
   * Note that this conversion can lose information about the
   * overall magnitude of the BigInteger value as well as return a
   * result with the opposite sign.
   *
   * @return this BigInteger converted to a {@code long}.
   * @see #longValueExact()
   */
  public long longValue() {
    long result = 0;

    for (int i = 1; i >= 0; i--) {
      result = (result << 32) + (getInt(i) & LONG_MASK);
    }
    return result;
  }

  /**
   * Converts this BigInteger to a {@code float}.  This
   * conversion is similar to the
   * <i>narrowing primitive conversion</i> from {@code double} to
   * {@code float} as defined in section 5.1.3 of
   * <cite>The Java&trade; Language Specification</cite>:
   * if this BigInteger has too great a magnitude
   * to represent as a {@code float}, it will be converted to
   * {@link Float#NEGATIVE_INFINITY} or {@link
   * Float#POSITIVE_INFINITY} as appropriate.  Note that even when
   * the return value is finite, this conversion can lose
   * information about the precision of the BigInteger value.
   *
   * @return this BigInteger converted to a {@code float}.
   */
  public float floatValue() {
    if (signum == 0) {
      return 0.0f;
    }

    int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;

    // exponent == floor(log2(abs(this)))
    if (exponent < Long.SIZE - 1) {
      return longValue();
    } else if (exponent > Float.MAX_EXPONENT) {
      return signum > 0 ? Float.POSITIVE_INFINITY : Float.NEGATIVE_INFINITY;
    }

        /*
         * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
         * one bit. To make rounding easier, we pick out the top
         * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
         * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
         * bits, and signifFloor the top SIGNIFICAND_WIDTH.
         *
         * It helps to consider the real number signif = abs(this) *
         * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
         */
    int shift = exponent - FloatConsts.SIGNIFICAND_WIDTH;

    int twiceSignifFloor;
    // twiceSignifFloor will be == abs().shiftRight(shift).intValue()
    // We do the shift into an int directly to improve performance.

    int nBits = shift & 0x1f;
    int nBits2 = 32 - nBits;

    if (nBits == 0) {
      twiceSignifFloor = mag[0];
    } else {
      twiceSignifFloor = mag[0] >>> nBits;
      if (twiceSignifFloor == 0) {
        twiceSignifFloor = (mag[0] << nBits2) | (mag[1] >>> nBits);
      }
    }

    int signifFloor = twiceSignifFloor >> 1;
    signifFloor &= FloatConsts.SIGNIF_BIT_MASK; // remove the implied bit

        /*
         * We round up if either the fractional part of signif is strictly
         * greater than 0.5 (which is true if the 0.5 bit is set and any lower
         * bit is set), or if the fractional part of signif is >= 0.5 and
         * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
         * are set). This is equivalent to the desired HALF_EVEN rounding.
         */
    boolean increment = (twiceSignifFloor & 1) != 0
        && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
    int signifRounded = increment ? signifFloor + 1 : signifFloor;
    int bits = ((exponent + FloatConsts.EXP_BIAS))
        << (FloatConsts.SIGNIFICAND_WIDTH - 1);
    bits += signifRounded;
        /*
         * If signifRounded == 2^24, we'd need to set all of the significand
         * bits to zero and add 1 to the exponent. This is exactly the behavior
         * we get from just adding signifRounded to bits directly. If the
         * exponent is Float.MAX_EXPONENT, we round up (correctly) to
         * Float.POSITIVE_INFINITY.
         */
    bits |= signum & FloatConsts.SIGN_BIT_MASK;
    return Float.intBitsToFloat(bits);
  }

  /**
   * Converts this BigInteger to a {@code double}.  This
   * conversion is similar to the
   * <i>narrowing primitive conversion</i> from {@code double} to
   * {@code float} as defined in section 5.1.3 of
   * <cite>The Java&trade; Language Specification</cite>:
   * if this BigInteger has too great a magnitude
   * to represent as a {@code double}, it will be converted to
   * {@link Double#NEGATIVE_INFINITY} or {@link
   * Double#POSITIVE_INFINITY} as appropriate.  Note that even when
   * the return value is finite, this conversion can lose
   * information about the precision of the BigInteger value.
   *
   * @return this BigInteger converted to a {@code double}.
   */
  public double doubleValue() {
    if (signum == 0) {
      return 0.0;
    }

    int exponent = ((mag.length - 1) << 5) + bitLengthForInt(mag[0]) - 1;

    // exponent == floor(log2(abs(this))Double)
    if (exponent < Long.SIZE - 1) {
      return longValue();
    } else if (exponent > Double.MAX_EXPONENT) {
      return signum > 0 ? Double.POSITIVE_INFINITY : Double.NEGATIVE_INFINITY;
    }

        /*
         * We need the top SIGNIFICAND_WIDTH bits, including the "implicit"
         * one bit. To make rounding easier, we pick out the top
         * SIGNIFICAND_WIDTH + 1 bits, so we have one to help us round up or
         * down. twiceSignifFloor will contain the top SIGNIFICAND_WIDTH + 1
         * bits, and signifFloor the top SIGNIFICAND_WIDTH.
         *
         * It helps to consider the real number signif = abs(this) *
         * 2^(SIGNIFICAND_WIDTH - 1 - exponent).
         */
    int shift = exponent - DoubleConsts.SIGNIFICAND_WIDTH;

    long twiceSignifFloor;
    // twiceSignifFloor will be == abs().shiftRight(shift).longValue()
    // We do the shift into a long directly to improve performance.

    int nBits = shift & 0x1f;
    int nBits2 = 32 - nBits;

    int highBits;
    int lowBits;
    if (nBits == 0) {
      highBits = mag[0];
      lowBits = mag[1];
    } else {
      highBits = mag[0] >>> nBits;
      lowBits = (mag[0] << nBits2) | (mag[1] >>> nBits);
      if (highBits == 0) {
        highBits = lowBits;
        lowBits = (mag[1] << nBits2) | (mag[2] >>> nBits);
      }
    }

    twiceSignifFloor = ((highBits & LONG_MASK) << 32)
        | (lowBits & LONG_MASK);

    long signifFloor = twiceSignifFloor >> 1;
    signifFloor &= DoubleConsts.SIGNIF_BIT_MASK; // remove the implied bit

        /*
         * We round up if either the fractional part of signif is strictly
         * greater than 0.5 (which is true if the 0.5 bit is set and any lower
         * bit is set), or if the fractional part of signif is >= 0.5 and
         * signifFloor is odd (which is true if both the 0.5 bit and the 1 bit
         * are set). This is equivalent to the desired HALF_EVEN rounding.
         */
    boolean increment = (twiceSignifFloor & 1) != 0
        && ((signifFloor & 1) != 0 || abs().getLowestSetBit() < shift);
    long signifRounded = increment ? signifFloor + 1 : signifFloor;
    long bits = (long) ((exponent + DoubleConsts.EXP_BIAS))
        << (DoubleConsts.SIGNIFICAND_WIDTH - 1);
    bits += signifRounded;
        /*
         * If signifRounded == 2^53, we'd need to set all of the significand
         * bits to zero and add 1 to the exponent. This is exactly the behavior
         * we get from just adding signifRounded to bits directly. If the
         * exponent is Double.MAX_EXPONENT, we round up (correctly) to
         * Double.POSITIVE_INFINITY.
         */
    bits |= signum & DoubleConsts.SIGN_BIT_MASK;
    return Double.longBitsToDouble(bits);
  }

  /**
   * Returns a copy of the input array stripped of any leading zero bytes.
   */
  private static int[] stripLeadingZeroInts(int val[]) {
    int vlen = val.length;
    int keep;

    // Find first nonzero byte
    for (keep = 0; keep < vlen && val[keep] == 0; keep++) {
      ;
    }
    return java.util.Arrays.copyOfRange(val, keep, vlen);
  }

  /**
   * Returns the input array stripped of any leading zero bytes.
   * Since the source is trusted the copying may be skipped.
   */
  private static int[] trustedStripLeadingZeroInts(int val[]) {
    int vlen = val.length;
    int keep;

    // Find first nonzero byte
    for (keep = 0; keep < vlen && val[keep] == 0; keep++) {
      ;
    }
    return keep == 0 ? val : java.util.Arrays.copyOfRange(val, keep, vlen);
  }

  /**
   * Returns a copy of the input array stripped of any leading zero bytes.
   */
  private static int[] stripLeadingZeroBytes(byte a[]) {
    int byteLength = a.length;
    int keep;

    // Find first nonzero byte
    for (keep = 0; keep < byteLength && a[keep] == 0; keep++) {
      ;
    }

    // Allocate new array and copy relevant part of input array
    int intLength = ((byteLength - keep) + 3) >>> 2;
    int[] result = new int[intLength];
    int b = byteLength - 1;
    for (int i = intLength - 1; i >= 0; i--) {
      result[i] = a[b--] & 0xff;
      int bytesRemaining = b - keep + 1;
      int bytesToTransfer = Math.min(3, bytesRemaining);
      for (int j = 8; j <= (bytesToTransfer << 3); j += 8) {
        result[i] |= ((a[b--] & 0xff) << j);
      }
    }
    return result;
  }

  /**
   * Takes an array a representing a negative 2's-complement number and
   * returns the minimal (no leading zero bytes) unsigned whose value is -a.
   */
  private static int[] makePositive(byte a[]) {
    int keep, k;
    int byteLength = a.length;

    // Find first non-sign (0xff) byte of input
    for (keep = 0; keep < byteLength && a[keep] == -1; keep++) {
      ;
    }


        /* Allocate output array.  If all non-sign bytes are 0x00, we must
         * allocate space for one extra output byte. */
    for (k = keep; k < byteLength && a[k] == 0; k++) {
      ;
    }

    int extraByte = (k == byteLength) ? 1 : 0;
    int intLength = ((byteLength - keep + extraByte) + 3) >>> 2;
    int result[] = new int[intLength];

        /* Copy one's complement of input into output, leaving extra
         * byte (if it exists) == 0x00 */
    int b = byteLength - 1;
    for (int i = intLength - 1; i >= 0; i--) {
      result[i] = a[b--] & 0xff;
      int numBytesToTransfer = Math.min(3, b - keep + 1);
      if (numBytesToTransfer < 0) {
        numBytesToTransfer = 0;
      }
      for (int j = 8; j <= 8 * numBytesToTransfer; j += 8) {
        result[i] |= ((a[b--] & 0xff) << j);
      }

      // Mask indicates which bits must be complemented
      int mask = -1 >>> (8 * (3 - numBytesToTransfer));
      result[i] = ~result[i] & mask;
    }

    // Add one to one's complement to generate two's complement
    for (int i = result.length - 1; i >= 0; i--) {
      result[i] = (int) ((result[i] & LONG_MASK) + 1);
      if (result[i] != 0) {
        break;
      }
    }

    return result;
  }

  /**
   * Takes an array a representing a negative 2's-complement number and
   * returns the minimal (no leading zero ints) unsigned whose value is -a.
   */
  private static int[] makePositive(int a[]) {
    int keep, j;

    // Find first non-sign (0xffffffff) int of input
    for (keep = 0; keep < a.length && a[keep] == -1; keep++) {
      ;
    }

        /* Allocate output array.  If all non-sign ints are 0x00, we must
         * allocate space for one extra output int. */
    for (j = keep; j < a.length && a[j] == 0; j++) {
      ;
    }
    int extraInt = (j == a.length ? 1 : 0);
    int result[] = new int[a.length - keep + extraInt];

        /* Copy one's complement of input into output, leaving extra
         * int (if it exists) == 0x00 */
    for (int i = keep; i < a.length; i++) {
      result[i - keep + extraInt] = ~a[i];
    }

    // Add one to one's complement to generate two's complement
    for (int i = result.length - 1; ++result[i] == 0; i--) {
      ;
    }

    return result;
  }

  /*
     * The following two arrays are used for fast String conversions.  Both
     * are indexed by radix.  The first is the number of digits of the given
     * radix that can fit in a Java long without "going negative", i.e., the
     * highest integer n such that radix**n < 2**63.  The second is the
     * "long radix" that tears each number into "long digits", each of which
     * consists of the number of digits in the corresponding element in
     * digitsPerLong (longRadix[i] = i**digitPerLong[i]).  Both arrays have
     * nonsense values in their 0 and 1 elements, as radixes 0 and 1 are not
     * used.
     */
  private static int digitsPerLong[] = {0, 0,
      62, 39, 31, 27, 24, 22, 20, 19, 18, 18, 17, 17, 16, 16, 15, 15, 15, 14,
      14, 14, 14, 13, 13, 13, 13, 13, 13, 12, 12, 12, 12, 12, 12, 12, 12};

  private static BigInteger longRadix[] = {null, null,
      valueOf(0x4000000000000000L), valueOf(0x383d9170b85ff80bL),
      valueOf(0x4000000000000000L), valueOf(0x6765c793fa10079dL),
      valueOf(0x41c21cb8e1000000L), valueOf(0x3642798750226111L),
      valueOf(0x1000000000000000L), valueOf(0x12bf307ae81ffd59L),
      valueOf(0xde0b6b3a7640000L), valueOf(0x4d28cb56c33fa539L),
      valueOf(0x1eca170c00000000L), valueOf(0x780c7372621bd74dL),
      valueOf(0x1e39a5057d810000L), valueOf(0x5b27ac993df97701L),
      valueOf(0x1000000000000000L), valueOf(0x27b95e997e21d9f1L),
      valueOf(0x5da0e1e53c5c8000L), valueOf(0xb16a458ef403f19L),
      valueOf(0x16bcc41e90000000L), valueOf(0x2d04b7fdd9c0ef49L),
      valueOf(0x5658597bcaa24000L), valueOf(0x6feb266931a75b7L),
      valueOf(0xc29e98000000000L), valueOf(0x14adf4b7320334b9L),
      valueOf(0x226ed36478bfa000L), valueOf(0x383d9170b85ff80bL),
      valueOf(0x5a3c23e39c000000L), valueOf(0x4e900abb53e6b71L),
      valueOf(0x7600ec618141000L), valueOf(0xaee5720ee830681L),
      valueOf(0x1000000000000000L), valueOf(0x172588ad4f5f0981L),
      valueOf(0x211e44f7d02c1000L), valueOf(0x2ee56725f06e5c71L),
      valueOf(0x41c21cb8e1000000L)};

  /*
     * These two arrays are the integer analogue of above.
     */
  private static int digitsPerInt[] = {0, 0, 30, 19, 15, 13, 11,
      11, 10, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6,
      6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5};

  private static int intRadix[] = {0, 0,
      0x40000000, 0x4546b3db, 0x40000000, 0x48c27395, 0x159fd800,
      0x75db9c97, 0x40000000, 0x17179149, 0x3b9aca00, 0xcc6db61,
      0x19a10000, 0x309f1021, 0x57f6c100, 0xa2f1b6f, 0x10000000,
      0x18754571, 0x247dbc80, 0x3547667b, 0x4c4b4000, 0x6b5a6e1d,
      0x6c20a40, 0x8d2d931, 0xb640000, 0xe8d4a51, 0x1269ae40,
      0x17179149, 0x1cb91000, 0x23744899, 0x2b73a840, 0x34e63b41,
      0x40000000, 0x4cfa3cc1, 0x5c13d840, 0x6d91b519, 0x39aa400
  };

  /**
   * These routines provide access to the two's complement representation
   * of BigIntegers.
   */

  /**
   * Returns the length of the two's complement representation in ints,
   * including space for at least one sign bit.
   */
  private int intLength() {
    return (bitLength() >>> 5) + 1;
  }

  /* Returns sign bit */
  private int signBit() {
    return signum < 0 ? 1 : 0;
  }

  /* Returns an int of sign bits */
  private int signInt() {
    return signum < 0 ? -1 : 0;
  }

  /**
   * Returns the specified int of the little-endian two's complement
   * representation (int 0 is the least significant).  The int number can
   * be arbitrarily high (values are logically preceded by infinitely many
   * sign ints).
   */
  private int getInt(int n) {
    if (n < 0) {
      return 0;
    }
    if (n >= mag.length) {
      return signInt();
    }

    int magInt = mag[mag.length - n - 1];

    return (signum >= 0 ? magInt :
        (n <= firstNonzeroIntNum() ? -magInt : ~magInt));
  }

  /**
   * Returns the index of the int that contains the first nonzero int in the
   * little-endian binary representation of the magnitude (int 0 is the
   * least significant). If the magnitude is zero, return value is undefined.
   */
  private int firstNonzeroIntNum() {
    int fn = firstNonzeroIntNum - 2;
    if (fn == -2) { // firstNonzeroIntNum not initialized yet
      fn = 0;

      // Search for the first nonzero int
      int i;
      int mlen = mag.length;
      for (i = mlen - 1; i >= 0 && mag[i] == 0; i--) {
        ;
      }
      fn = mlen - i - 1;
      firstNonzeroIntNum = fn + 2; // offset by two to initialize
    }
    return fn;
  }

  /**
   * use serialVersionUID from JDK 1.1. for interoperability
   */
  private static final long serialVersionUID = -8287574255936472291L;

  /**
   * Serializable fields for BigInteger.
   *
   * @serialField signum  int signum of this BigInteger.
   * @serialField magnitude int[] magnitude array of this BigInteger.
   * @serialField bitCount  int number of bits in this BigInteger
   * @serialField bitLength int the number of bits in the minimal two's-complement representation of
   * this BigInteger
   * @serialField lowestSetBit int lowest set bit in the twos complement representation
   */
  private static final ObjectStreamField[] serialPersistentFields = {
      new ObjectStreamField("signum", Integer.TYPE),
      new ObjectStreamField("magnitude", byte[].class),
      new ObjectStreamField("bitCount", Integer.TYPE),
      new ObjectStreamField("bitLength", Integer.TYPE),
      new ObjectStreamField("firstNonzeroByteNum", Integer.TYPE),
      new ObjectStreamField("lowestSetBit", Integer.TYPE)
  };

  /**
   * Reconstitute the {@code BigInteger} instance from a stream (that is,
   * deserialize it). The magnitude is read in as an array of bytes
   * for historical reasons, but it is converted to an array of ints
   * and the byte array is discarded.
   * Note:
   * The current convention is to initialize the cache fields, bitCount,
   * bitLength and lowestSetBit, to 0 rather than some other marker value.
   * Therefore, no explicit action to set these fields needs to be taken in
   * readObject because those fields already have a 0 value be default since
   * defaultReadObject is not being used.
   */
  private void readObject(java.io.ObjectInputStream s)
      throws java.io.IOException, ClassNotFoundException {
        /*
         * In order to maintain compatibility with previous serialized forms,
         * the magnitude of a BigInteger is serialized as an array of bytes.
         * The magnitude field is used as a temporary store for the byte array
         * that is deserialized. The cached computation fields should be
         * transient but are serialized for compatibility reasons.
         */

    // prepare to read the alternate persistent fields
    ObjectInputStream.GetField fields = s.readFields();

    // Read the alternate persistent fields that we care about
    int sign = fields.get("signum", -2);
    byte[] magnitude = (byte[]) fields.get("magnitude", null);

    // Validate signum
    if (sign < -1 || sign > 1) {
      String message = "BigInteger: Invalid signum value";
      if (fields.defaulted("signum")) {
        message = "BigInteger: Signum not present in stream";
      }
      throw new java.io.StreamCorruptedException(message);
    }
    int[] mag = stripLeadingZeroBytes(magnitude);
    if ((mag.length == 0) != (sign == 0)) {
      String message = "BigInteger: signum-magnitude mismatch";
      if (fields.defaulted("magnitude")) {
        message = "BigInteger: Magnitude not present in stream";
      }
      throw new java.io.StreamCorruptedException(message);
    }

    // Commit final fields via Unsafe
    UnsafeHolder.putSign(this, sign);

    // Calculate mag field from magnitude and discard magnitude
    UnsafeHolder.putMag(this, mag);
    if (mag.length >= MAX_MAG_LENGTH) {
      try {
        checkRange();
      } catch (ArithmeticException e) {
        throw new java.io.StreamCorruptedException("BigInteger: Out of the supported range");
      }
    }
  }

  // Support for resetting final fields while deserializing
  private static class UnsafeHolder {

    private static final sun.misc.Unsafe unsafe;
    private static final long signumOffset;
    private static final long magOffset;

    static {
      try {
        unsafe = sun.misc.Unsafe.getUnsafe();
        signumOffset = unsafe.objectFieldOffset
            (BigInteger.class.getDeclaredField("signum"));
        magOffset = unsafe.objectFieldOffset
            (BigInteger.class.getDeclaredField("mag"));
      } catch (Exception ex) {
        throw new ExceptionInInitializerError(ex);
      }
    }

    static void putSign(BigInteger bi, int sign) {
      unsafe.putIntVolatile(bi, signumOffset, sign);
    }

    static void putMag(BigInteger bi, int[] magnitude) {
      unsafe.putObjectVolatile(bi, magOffset, magnitude);
    }
  }

  /**
   * Save the {@code BigInteger} instance to a stream.
   * The magnitude of a BigInteger is serialized as a byte array for
   * historical reasons.
   *
   * @serialData two necessary fields are written as well as obsolete fields for compatibility with
   * older versions.
   */
  private void writeObject(ObjectOutputStream s) throws IOException {
    // set the values of the Serializable fields
    ObjectOutputStream.PutField fields = s.putFields();
    fields.put("signum", signum);
    fields.put("magnitude", magSerializedForm());
    // The values written for cached fields are compatible with older
    // versions, but are ignored in readObject so don't otherwise matter.
    fields.put("bitCount", -1);
    fields.put("bitLength", -1);
    fields.put("lowestSetBit", -2);
    fields.put("firstNonzeroByteNum", -2);

    // save them
    s.writeFields();
  }

  /**
   * Returns the mag array as an array of bytes.
   */
  private byte[] magSerializedForm() {
    int len = mag.length;

    int bitLen = (len == 0 ? 0 : ((len - 1) << 5) + bitLengthForInt(mag[0]));
    int byteLen = (bitLen + 7) >>> 3;
    byte[] result = new byte[byteLen];

    for (int i = byteLen - 1, bytesCopied = 4, intIndex = len - 1, nextInt = 0;
        i >= 0; i--) {
      if (bytesCopied == 4) {
        nextInt = mag[intIndex--];
        bytesCopied = 1;
      } else {
        nextInt >>>= 8;
        bytesCopied++;
      }
      result[i] = (byte) nextInt;
    }
    return result;
  }

  /**
   * Converts this {@code BigInteger} to a {@code long}, checking
   * for lost information.  If the value of this {@code BigInteger}
   * is out of the range of the {@code long} type, then an
   * {@code ArithmeticException} is thrown.
   *
   * @return this {@code BigInteger} converted to a {@code long}.
   * @throws ArithmeticException if the value of {@code this} will not exactly fit in a {@code
   * long}.
   * @see BigInteger#longValue
   * @since 1.8
   */
  public long longValueExact() {
    if (mag.length <= 2 && bitLength() <= 63) {
      return longValue();
    } else {
      throw new ArithmeticException("BigInteger out of long range");
    }
  }

  /**
   * Converts this {@code BigInteger} to an {@code int}, checking
   * for lost information.  If the value of this {@code BigInteger}
   * is out of the range of the {@code int} type, then an
   * {@code ArithmeticException} is thrown.
   *
   * @return this {@code BigInteger} converted to an {@code int}.
   * @throws ArithmeticException if the value of {@code this} will not exactly fit in a {@code
   * int}.
   * @see BigInteger#intValue
   * @since 1.8
   */
  public int intValueExact() {
    if (mag.length <= 1 && bitLength() <= 31) {
      return intValue();
    } else {
      throw new ArithmeticException("BigInteger out of int range");
    }
  }

  /**
   * Converts this {@code BigInteger} to a {@code short}, checking
   * for lost information.  If the value of this {@code BigInteger}
   * is out of the range of the {@code short} type, then an
   * {@code ArithmeticException} is thrown.
   *
   * @return this {@code BigInteger} converted to a {@code short}.
   * @throws ArithmeticException if the value of {@code this} will not exactly fit in a {@code
   * short}.
   * @see BigInteger#shortValue
   * @since 1.8
   */
  public short shortValueExact() {
    if (mag.length <= 1 && bitLength() <= 31) {
      int value = intValue();
      if (value >= Short.MIN_VALUE && value <= Short.MAX_VALUE) {
        return shortValue();
      }
    }
    throw new ArithmeticException("BigInteger out of short range");
  }

  /**
   * Converts this {@code BigInteger} to a {@code byte}, checking
   * for lost information.  If the value of this {@code BigInteger}
   * is out of the range of the {@code byte} type, then an
   * {@code ArithmeticException} is thrown.
   *
   * @return this {@code BigInteger} converted to a {@code byte}.
   * @throws ArithmeticException if the value of {@code this} will not exactly fit in a {@code
   * byte}.
   * @see BigInteger#byteValue
   * @since 1.8
   */
  public byte byteValueExact() {
    if (mag.length <= 1 && bitLength() <= 31) {
      int value = intValue();
      if (value >= Byte.MIN_VALUE && value <= Byte.MAX_VALUE) {
        return byteValue();
      }
    }
    throw new ArithmeticException("BigInteger out of byte range");
  }
}
